DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY

c 2001 by CRC Press LLC Comprehensive Dictionary of Mathematics

Douglas N. Clark Editor-in-Chief Stan Gibilisco Editorial Advisor

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Analysis, Calculus, and Differential Equations Douglas N. Clark Algebra, Arithmetic and Trigonometry Steven G. Krantz

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c 2001 by CRC Press LLC a Volume in the Comprehensive Dictionary of Mathematics

DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY

Edited by Steven G. Krantz

CRC Press Boca Raton London New York Washington, D.C.

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The second volume of the CRC Press Comprehensive Dictionary of Mathematics covers algebra, arithmetic and trigonometry broadly, with an overlap into differential geometry, algebraic geometry, topology and other related fields. The authorship is by well over 30 mathematicians, active in teaching and research, including the editor. Because it is a dictionary and not an encyclopedia, definitions are only occasionally accompanied by a discussion or example. In a dictionary of mathematics, the primary goal is to define each term rigorously. The derivation of a term is almost never attempted. The dictionary is written to be a useful reference for a readership that includes students, scientists, and engineers with a wide range of backgrounds, as well as specialists in areas of analysis and differential equations and mathematicians in related fields. Therefore, the definitions are intended to be accessible, as well as rigorous. To be sure, the degree of accessibility may depend upon the individual term, in a dictionary with terms ranging from Abelian cohomology to z intercept. Occasionally a term must be omitted because it is archaic. Care was taken when such circum- stances arose to ensure that the term was obsolete. An example of an archaic term deemed to be obsolete, and hence not included, is “right line”. This term was used throughout a turn-of-the-century analytic geometry textbook we needed to consult, but it was not defined there. Finally, reference to a contemporary English language dictionary yielded “straight line” as a synonym for “right line”. The authors are grateful to the series editor, Stanley Gibilisco, for dealing with our seemingly endless procedural questions and to Nora Konopka, for always acting efficiently and cheerfully with CRC Press liaison matters.

Douglas N. Clark Editor-in-Chief

c 2001 by CRC Press LLC CONTRIBUTORS

Edward Aboufadel Neil K. Dickson Grand Valley State University University of Glasgow Allendale, Michigan Glasgow, United Kingdom

Gerardo Aladro David E. Dobbs Florida International University University of Tennessee Miami, Florida Knoxville, Tennessee

Mohammad Azarian Marcus Feldman University of Evansville Washington University Evansville. Indiana St. Louis, Missouri

Susan Barton Stephen Humphries West Virginia Institute of Technology Brigham Young University Montgomery, West Virginia Provo, Utah

Albert Boggess Shanyu Ji Texas A&M University University of Houston College Station, Texas Houston, Texas

Robert Borrelli Kenneth D. Johnson Harvey Mudd College University of Georgia Claremont, California Athens, Georgia

Stephen W. Brady Bao Qin Li Wichita State University Florida International University Wichita, Kansas Miami, Florida

Der Chen Chang Robert E. MacRae Georgetown University University of Colorado Washington, D.C. Boulder, Colorado

Stephen A. Chiappari Charles N. Moore Santa Clara University Kansas State University Santa Clara. California Manhattan, Kansas

Joseph A. Cima Hossein Movahedi-Lankarani The University of North Carolina at Chapel Hill Pennsylvania State University Chapel Hill, North Carolina Altoona, Pennsylvania

Courtney S. Coleman Shashikant B. Mulay Harvey Mudd College University of Tennessee Claremont, California Knoxville, Tennessee

John B. Conway Judy Kenney Munshower University of Tennessee Avila College Knoxville, Tennessee Kansas City, Missouri

c 2001 by CRC Press LLC Charles W. Neville Anthony D. Thomas CWN Research University of Wisconsin Berlin, Connecticut Platteville. Wisconsin

Daniel E. Otero Michael Tsatsomeros Xavier University University of Regina Cincinnati, Ohio Regina, Saskatchewan,Canada

Josef Paldus James S. Walker University of Waterloo University of Wisconsin at Eau Claire Waterloo, Ontario, Canada Eau Claire, Wisconsin

Harold R. Parks C. Eugene Wayne Oregon State University Boston University Corvallis, Oregon Boston, Massachusetts

Gunnar Stefansson Kehe Zhu Pennsylvania State University State University of New York at Albany Altoona, Pennsylvania Albany, New York

c 2001 by CRC Press LLC its Galois group is an Abelian group. See Galois group. See also Abelian group.

Abelian extension A Galois extension of a A field is called an Abelian extension if its Galois group is Abelian. See Galois extension. See also Abelian group. A-balanced mapping Let M be a right module over the ring A, and let N be a left module Abelian function A function f(z1,z2,z3, n over the same ring A. A mapping φ from M ×N ...,zn) meromorphic on C for which there ex- n to an Abelian group G is said to be A-balanced ist 2n vectors ωk ∈ C , k = 1, 2, 3,...,2n, if φ(x,·) is a group homomorphism from N to called period vectors, that are linearly indepen- G for each x ∈ M,ifφ(·,y)is a group homo- dent over R and are such that morphism from M to G for each y ∈ N, and + = if f (z ωk) f(z) φ(xa,y) = φ(x,ay) holds for k = 1, 2, 3,...,2n and z ∈ Cn. holds for all x ∈ M, y ∈ N, and a ∈ A. Abelian function field The set of Abelian A-B-bimodule An Abelian group G that is a functions on Cn corresponding to a given set of left module over the ring A and a right module period vectors forms a field called an Abelian over the ring B and satisfies the associative law function field. (ax)b = a(xb) for all a ∈ A, b ∈ B, and all x ∈ G. Abelian group Briefly, a commutative group. More completely, a set G, together with a binary Abelian cohomology The usual cohomology operation, usually denoted “+,” a unary opera- with coefficients in an Abelian group; used if tion usually denoted “−,” and a distinguished the context requires one to distinguish between element usually denoted “0” satisfying the fol- the usual cohomology and the more exotic non- lowing axioms: Abelian cohomology. See cohomology. (i.) a + (b + c) = (a + b) + c for all a,b,c ∈ G, Abelian differential of the first kind A holo- (ii.) a + 0 = a for all a ∈ G, morphic differential on a closed Riemann sur- (iii.) a + (−a) = 0 for all a ∈ G, face; that is, a differential of the form ω = (iv.) a + b = b + a for all a,b ∈ G. a(z)dz, where a(z) is a holomorphic function. The element 0 is called the identity, −a is called the inverse of a, axiom (i.) is called the Abelian differential of the second kind A associative axiom, and axiom (iv.) is called the meromorphic differential on a closed Riemann commutative axiom. surface, the singularities of which are all of order greater than or equal to 2; that is, a differential Abelian ideal An ideal in a Lie algebra which of the form ω = a(z)dz where a(z) is a mero- forms a commutative subalgebra. morphic function with only 0 residues. Abelian integral of the first kind An indef- Abelian differential of the third kind A inite integral W(p) = p a(z)dz on a closed p0 differential on a closed Riemann surface that is Riemann surface in which the function a(z) is not an Abelian differential of the first or sec- holomorphic (the differential ω(z) = a(z)dz ond kind; that is, a differential of the form ω = is said to be an Abelian differential of the first a(z)dz where a(z) is meromorphic and has at kind). least one non-zero residue. Abelian integral of the second kind An in- Abelian equation A polynomial equation definite integral W(p) = p a(z)dzon a closed p0 f(X) = 0 is said to be an Abelian equation if Riemann surface in which the function a(z) is

c 2001 by CRC Press LLC meromorphic with all its singularities of order as the product of prime ideals of K of absolute at least 2 (the differential a(z)dz is said to be an degree 1 if and only if p is a principal ideal. Abelian differential of the second kind). The term “absolute class field” is used to distinguish the Galois extensions described above, Abelian integral of the third kind An in- which were introduced by Hilbert, from a more definite integral W(p) = p a(z)dzon a closed p0 general concept of “class field” defined by Riemann surface in which the function a(z) is Tagaki. See also class field. meromorphic and has at least one non-zero residue (the differential a(z)dzis said to be an Abel- absolute covariant A covariant of weight 0. ian differential of the third kind). See also covariant.

Abelian Lie group A Lie group for which absolute inequality An inequality involving the associated Lie algebra is Abelian. See also variables that is valid for all possible substitu- Lie algebra. tions of real numbers for the variables. Abelian projection operator A non-zero absolute invariant Any quantity or property projection operator E in a von Neumann algebra of an algebraic variety that is preserved under M such that the reduced von Neumann algebra birational transformations. ME = EME is Abelian.

Abelian subvariety A subvariety of an absolutely irreducible character The char- Abelian variety that is also a subgroup. See also acter of an absolutely irreducible representation. Abelian variety. A representation is absolutely irreducible if it is irreducible and if the representation obtained by making an extension of the ground ﬁeld remains Abelian surface A two-dimensional Abelian irreducible. variety. See also Abelian variety.

Abelian variety A complete algebraic vari- absolutely irreducible representation A ety G that also forms a commutative algebraic representation is absolutely irreducible if it is group. That is, G is a group under group oper- irreducible and if the representation obtained by ations that are regular functions. The fact that making an extension of the ground field remains an algebraic group is complete as an algebraic irreducible. variety implies that the group is commutative. See also regular function. absolutely simple group A group that contains no serial subgroup. The notion of an ab- Abel’s Theorem Niels Henrik Abel (1802- solutely simple group is a strengthening of the 1829) proved several results now known as concept of a simple group that is appropriate for “Abel’s Theorem,” but perhaps preeminent infinite groups. See serial subgroup. among these is Abel’s proof that the general quintic equation cannot be solved algebraically. absolutely uniserial algebra Let A be an al- Other theorems that may be found under the gebra over the field K, and let L be an extension heading “Abel’s Theorem” concern power se- field of K. Then L ⊗K A can be regarded as ries, Dirichlet series, and divisors on Riemann an algebra over L. If, for every choice of L, surfaces. L ⊗K A can be decomposed into a direct sum of ideals which are primary rings, then A is an absolute class field Let k be an algebraic absolutely uniserial algebra. number field. A Galois extension K of k is an absolute class field if it satisfies the following absolute multiple covariant A multiple co- property regarding prime ideals of k: A prime variant of weight 0. See also multiple covari- ideal p of k of absolute degree 1 decomposes ants.

c 2001 by CRC Press LLC absolute number A specific number repre- acceleration parameter A parameter chosen 3 sented by numerals such as 2, 4 , or 5.67 in con- in applying successive over-relaxation (which trast with a literal number which is a number is an accelerated version of the Gauss-Seidel represented by a letter. method) to solve a system of linear equations numerically. More specifically, one solves Ax = b absolute value of a complex number More iteratively by setting commonly called the modulus, the absolute val- = + ue of the complex number z a ib, where a xn+1 = xn + R (b − Axn) , and b are real, is denoted√ by |z| and equals the non-negative real number a2 + b2. where −1 −1 absolute value of a vector More commonly R = L + ω D called the magnitude, the absolute value of the vector with L the lower triangular submatrix of A, D −→ the diagonal of A, and 0 <ω<2. Here, ω v = (v1,v2,...,vn) is the acceleration parameter, also called the |−→| is denoted by v and equals the non-negative relaxation parameter. Analysis is required to 2 + 2 +···+ 2 real number v1 v2 vn. choose an appropriate value of ω. absolute value of real number For a real acyclic chain complex An augmented, pos- number r, the nonnegative real number |r|,given itive chain complex by r if r ≥ 0 ···−→∂n+1 −→∂n −→∂n−1 |r|= Xn Xn−1 ... −r if r<0 . ∂2 ∂1 ···−→ X1 −→ X0 → A → 0 abstract algebraic variety A set that is anal- forming an exact sequence. This in turn means ogous to an ordinary algebraic variety, but de- that the kernel of ∂ equals the image of ∂ + fined only locally and without an imbedding. n n 1 for n ≥ 1, the kernel of equals the image of ∂ X A abstract function (1) In the theory of gen- 1, and is surjective. Here the i and are eralized almost-periodic functions, a function modules over a commutative unitary ring. mapping R to a Banach space other than the complex numbers. addend In arithmetic, a number that is to be (2) A function from one Banach space to an- added to another number. In general, one of the other Banach space that is everywhere differen- operands of an operation of addition. See also tiable in the sense of Fréchet. addition. abstract variety A generalization of the no- addition (1) A basic arithmetic operation tion of an algebraic variety introduced by Weil, that expresses the relationship between the in analogy with the definition of a differentiable number of elements in each of two disjoint sets manifold. An abstract variety (also called an and the number of elements in the union of those abstract algebraic variety) consists of (i.) a two sets. family {Vα}α∈A of affine algebraic sets over a (2) The name of the binary operation in an given field k, (ii.) for each α ∈ A a family of Abelian group, when the notation “+” is used open subsets {Wαβ }β∈A of Vα, and (iii.) for each for that operation. See also Abelian group. pair α and β in A a birational transformation be- (3) The name of the binary operation in a tween Wαβ and Wαβ such that the composition ring, under which the elements form an Abelian of the birational transformations between subgroup. See also Abelian group. sets of Vα and Vβ and between subsets of Vβ (4) Sometimes, the name of one of the opera- and Vγ are consistent with those between sub- tions in a multi-operator group, even though the sets of Vα and Vγ . operation is not commutative.

c 2001 by CRC Press LLC addition formulas in trigonometry The for- for all s ∈ S. Such an additive identity is nec- mulas essarily unique and usually is denoted by “0.” cos(φ + θ) = cos φ cos θ − sin φ sin θ, In ordinary arithmetic, the number 0 is the additive identity because 0 + n = n + 0 = n sin(φ + θ) = cos φ sin θ + sin φ cos θ, holds for all numbers n. tan φ + tan θ (φ + θ) = . tan − 1 tan φ tan θ additive inverse In any algebraic structure with a commutative operation referred to as ad- addition of algebraic expressions One of dition and denoted by “+,” for which there is the fundamental ways of forming new algebraic an additive identity 0, the additive inverse of an expressions from existing algebraic expressions; element a is the element b for which a + b = the other methods of forming new expressions b + a = 0. The additive inverse of a is usu- from old being subtraction, multiplication, divi- ally denoted by −a. In arithmetic, the additive sion, and root extraction. inverse of a number is also called its opposite. See additive identity. addition of angles In elementary geometry or trigonometry, the angle resulting from the additive set function Let X be a set and let A process of following rotation through one an- be a collection of subsets of X that is closed un- gle about a center by rotation through another der the union operation. Let φ : A → F , where angle about the same center. F is a field of scalars. We say that φ is finitely ∈ A additive if, whenever S1,...,Sk are pair- addition of complex numbers One of the ∪k = k wise disjoint then φ( j=1Sj ) j=1 φ(Sj ). fundamental operations under which the com- We say that φ is countably additive if, when- plex numbers C form a field. If w = a + ib, ··· ∈ A ever S1,S2, are pairwise disjoint then z = c + id ∈ C, with a, b, c, and d real, then ∞ ∞ φ(∪ = Sj ) = = φ(Sj ). w + z = (a + c)+ i(b+ d) is the result of addi- j 1 j 1 tion, or the sum, of those two complex numbers. additive valuation Let F be a field and G be a totally ordered additive group. An addi- addition of vectors One of the fundamental tive valuation is a function v : F → G ∪ {∞} operations in a vector space, under which the set satisfying of vectors form an Abelian group. For vectors n n (i.) v(a) =∞if and only if a = 0, in R or C ,ifx = (x1,x2,...,xn) and y = = + (y1,y2,...,yn), then x + y = (x1 + y1,x2 + (ii.) v(ab) v(a) v(b), y2,...,xn + yn). (iii.) v(a + b) ≥ min{v(a), v(b)}. additive group (1) Any group, usually adele Following Weil, let k be either a finite Abelian, where the operation is denoted +. See algebraic extension of Q or a finitely generated group, Abelian group. extension of a finite prime field of transcendency (2) In discussing a ring R, the commutative degree 1 over that field. By a place of k is meant group formed by the elements of R under the the completion of the image of an isomorphic addition operation. embedding of k into a local field (actually the equivalence class of such completions under the additive identity In an Abelian group G, the equivalence relation induced by isomorphisms unique element (usually denoted 0) such that of the local fields). A place is infinite if the local g + 0 = g for all g ∈ G. field is R or C, otherwise the place is finite. For a place v, kv will denote the completion, and if additive identity a binary operation that is v is a finite place, rv will denote the maximal called addition and is denoted by “+.” In this compact subring of k .Anadele is an element ∈ v situation, an additive identity is an element i S of that satisfies the equation k × r , + = + = v v i s s i s v∈P v/∈P

c 2001 by CRC Press LLC where P is a finite set of places containing the the adjoint representation into the space of linear infinite places. endomorphisms of g. See also adjoint representation. adele group Let V be the set of valuations on the global field k.Forv ∈ V , let kv be adjoint matrix For a matrix M with complex the completion of k with respect to v, and let entries, the adjoint of M is denoted by M∗ and Ov be the ring of integer elements in kv. The is the complex conjugate of the transpose of M; ∗ adele group of the linear algebraic group G is so if M = mij , then M has m¯ ji as the entry the restricted direct product in its ith row and jth column. Gkv GOv adjoint representation (1) In the context of v∈V Lie algebras, the adjoint representation is the mapping sending X to [X, ·]. which, as a set, consists of all sequences of el- (2) In the context of Lie groups, the adjoint ements of G , indexed by v ∈ V , with all but kv representation is the mapping sending σ to the finitely many terms in each sequence being ele- differential of the automorphism ασ : G → G ments of GOv . −1 defined by ασ (τ) = στσ . adele ring Following Weil, let k be either a (3) In the context of representations of an al- finite algebraic extension of Q or a finitely gen- gebra over a field, the term adjoint representa- erated extension of a finite prime field of tran- tion is a synonym for dual representation. See scendency degree 1 over that field. Set dual representation. adjoint system Let D be a curve on a non- kA(P ) = kv × rv , singular surface S. The adjoint system of D is v∈P v/∈P |D + K|, where K is a canonical divisor on S. where P is a finite set of places of k containing the infinite places. A ring structure is put adjunction formula The formula on kA(P ) defining addition and multiplication componentwise. The adele ring is 2g − 2 = C.(C + K) kA = kA(P ) . relating the genus g of a non-singular curve C P on a surface S with the intersection pairing of C and C + K, where K is a canonical divisor on A locally compact topology is defined on kA by S. requiring each kA(P ) to be an open subring and using the product topology on kA(P ). admissible homomorphism For a group G with a set of operators , a group homomor- adjoining (1) Assuming K is a field exten- phism from G to a group G on which the same ⊂ sion of k and S K, the field obtained by ad- operators act, such that joining S to k is the smallest field F satisfying k ⊂ F ⊂ K and containing S. ω(ab) = (ωa)(ωb) (2)IfR is a commutative ring, then the ring of polynomials R[X] is said to be obtained by holds for all a,b ∈ G and all ω ∈ . Also called adjoining X to R. an -homomorphism or an operator homomorphism. adjoint group The image of a Lie group G, under the adjoint representation into the space admissible isomorphism For a group G with of linear endomorphisms of the associated Lie a set of operators , a group isomorphism from algebra g. See also adjoint representation. G onto a group G , on which the same operators act, such that adjoint Lie algebra Let g be a Lie algebra. The adjoint Lie algebra is the image of g under ω(ab) = (ωa)(ωb)

c 2001 by CRC Press LLC holds for all a,b ∈ G and all ω ∈ . Also called affectless equation A polynomial equation an -isomorphism or an operator isomorphism. for which the Galois group consists of all permutations. See also affect. admissible normal subgroup Let G be a group. It is easily seen that a subset N of G is affine algebraic group See linear algebraic a normal subgroup if and only if there is some group. equivalence relation ∼ on G such that ∼ is compatible with the multiplication on G, meaning affine morphism of schemes Let X and Y be schemes and f : X → Y be a morphism. If a ∼ b, c ∼ d ⇒ (ac) ∼ (bd) , there is an open affine cover {Vi} of the scheme −1 Y for which f (Vi) is affine for each i, then f and N is the equivalence class of the identity. is an affine morphism of schemes. In case G also has an operator domain ,an admissible normal subgroup is defined to be the affine scheme Let A be a commutative ring, equivalence class of the identity for an equiva- and let Spec(A) = X be the set of all prime lence relation ∼ that is compatible with the mul- ideals of A, equipped with the spectral or Zariski O tiplication as above and that also satisfies topology. Let X be a sheaf of local rings on X. The ringed space (X, OX) is called the affine a ∼ b ⇒ (ωa) ∼ (ωb) for all ω ∈ . scheme of the ring A.

affine space Let V be a real, linear n-dimen- admissible representation Let π be a uni- sional space. Let A be a set of points, which are tary representation of the group G in a Hilbert denoted P,Q. Define a relation between points space, and let M be the von Neumann algebra in A and vectors in V as follows: (i.) To every ordered pair (P, Q) ∈ A×A, there generated by π(G). The representation π is said −→ to be an admissible representation or a trace ad- is associated a “difference vector” PQ∈ V. missible representation if there exists a trace on (ii.) To every point P ∈ A and every vector + v ∈ V there is associated precisely one point M which is a character for π. −→ Q ∈ A such that PQ= v. Ado-Iwasawa Theorem The theorem that (iii.) If P, Q, R ∈ A then every finite dimensional Lie algebra (over a field −→ −→ −→ of characteristic p) has a faithful finite dimen- PQ+ QR = PR . sional representation. The characteristic p = 0 case of this is Ado’s Theorem and the charac- In this circumstance, we call A an n-dimensional teristic p = 0 case is Iwasawa’s Theorem. See affine space. also Lie algebra. affine variety A variety (common zero set Ado’s Theorem A finite dimensional Lie al- of a finite collection of functions) defined in an gebra g has a representation of finite degree ρ affine space. such that g =∼ ρ(g). A-homomorphism For A-modules M and While originally proved for Lie algebras N, a group homomorphism f : M → N is over fields of characteristic 0, the result was called an A-homomorphism if extended to characteristic p by Iwasawa. See Ado-Iwasawa Theorem. f(am)= af (m) for all a ∈ A, m ∈ M. affect For a polynomial equation P(X)= 0, the Galois group of the equation can be consid- Albanese variety For V a variety, the Al- ered as a group of permutations of the roots of banese variety of V is an Abelian variety A = the equation. The affect of the equation is the Alb(V ) such that there exists a rational f : index of the Galois group in the group of all V → A which generates A and has the uni- permutations of the roots of the equation. versal mapping property that for any rational

c 2001 by CRC Press LLC g : V → B, where B is an Abelian variety, In this case, M is called the kernel of the ex- there exist a homomorphism h : A → B and a tension because it is the kernel of the canonical constant c ∈ B such that g = hf + c. homomorphism.

Alexander Duality If A is a compact subset algebra homomorphism Suppose A and B of Rn, then for all indices q and all R-modules are algebras of the same type, meaning that for G, each n-ary operation fA on A there is a corresponding n-ary operation fB on B. A mapping n n n−q−1 H q (R , R \ A; G) = H (A; G) . φ : A → B is called a homomorphism from A to B if, for each pair of corresponding operations fA and fB , algebra (1) The system of symbolic ma- nipulation formalized by François Viéte (1540Ð φ (fA (a1,a2,...,an)) 1603), which today is known as elementary al- = f (φ (a ) ,φ(a ) ,...,φ(a )) gebra. B 1 2 n (2) The entire area of mathematics in which holds for all a ,a ,...,a ∈ A. one studies groups, rings, fields, etc. 1 2 n (3) A vector space (over a field) on which is Typically, an algebra A is a ring that also has also defined an operation of multiplication. the structure of a module over another ring R,so (4) A synonym for universal algebra, which that an algebra homomorphism φ must satisfy + = + ∈ includes structures such as Boolean algebras. (i.) φ(a1 a2) φ(a1) φ(a2) for a1,a2 A, (ii.) φ(a1a2) = φ(a1)φ(a2) for a1,a2 ∈ A, algebra class An equivalence class of central (iii.) φ(ra) = rφ(a), for r ∈ R and a ∈ A. simple algebras under the relation that relates a pair of algebras if they are both isomorphic to algebraic (1) An adjective referring to an full matrix rings over the same division algebra. object, structure, or theory that occurs in algebra Algebras in the same algebra class are said to be or arises through application of the processes “similar.” See also central simple algebra. used in algebra. (2) An adverb meaning a process that in- algebra class group Let K be a field. Two volves only the operations of algebra, which are central simple algebras over K are said to be addition, subtraction, multiplication, division, similar if they are isomorphic to full matrix rings and root extraction. over the same division algebra. Similarity is an equivalence relation, and the equivalence algebraic addition In elementary algebra, classes are called algebra classes. The product the addition of algebraic expressions which ex- of a pair of algebra classes is defined by choos- tends the operation of addition of numbers in ing an algebra from each class, say A and B, and arithmetic. letting the product of the classes be the algebra class containing A ⊗K B. This product is well algebraic addition formula For an Abelian defined, and the algebra classes form a group un- function f , an equation that expresses f(a+ b) der this multiplication, called the algebra class rationally, in terms of the values of a certain group or Brauer group. (p + 1)-tuple of Abelian functions, evaluated at the points a,b ∈ C. See also Abelian function. algebra extension Let A be an algebra over the commutative ring R. Then by an algebra algebraic algebra An algebra A over a field extension of A is meant either K such that every a ∈ A is algebraic over K. (i.) an algebra over R that contains A;or See algebra. (ii.) an algebra A containing a two-sided R- module M which is a two-sided ideal in A and algebraically closed field A field k, in which is such that every polynomial in one variable, with coeffi-

A /M = A. cients in k, has a root.

c 2001 by CRC Press LLC algebraic closure The smallest algebraically element of K if it satisfies a non-trivial polyno- closed extension field of a given field F . The mial equation with coefficients in k. algebraic closure exists and is unique up to isomorphism. algebraic equation An equation of the form P = 0 where P is a polynomial in one or more algebraic correspondence Let C be a non- variables. singular algebraic curve. By an algebraic correspondence is meant a divisor in the product algebraic equivalence Two cycles X1 and variety C × C. More generally, an algebraic X2 in a non-singular algebraic variety V are al- correspondence means a Zariski closed subset gebraically equivalent if there is a family of cy- T of the product V1 × V2 of two irreducible va- cles {X(t) : t ∈ T } on V , parameterized by rieties. Points P1 ∈ V1 and P2 ∈ V2 are said t ∈ T , where T is another non-singular alge- to correspond if (P1,P2) ∈ T . See also corre- braic variety, such that there is a cycle Z in V ×T spondence ring. for which each X(t) is the projection to V of the intersection of Z and V ×{t}, and X1 = X(t1), algebraic curve An algebraic variety of di- X2 = X(t2), for some t1,t2 ∈ T . Such a family mension one. See also algebraic variety. of cycles X(t) is called an algebraic family. algebraic cycle By an algebraic cycle of di- algebraic equivalence of divisors Two dimension m on an algebraic variety V is meant a visors f and g on an irreducible variety X are finite formal sum algebraically equivalent if there exists an alge- ∈ c V braic family of divisors, ft , t T , and points i i ∈ = = t1 and t2 T , such that f ft1 , and g ft2 . where the ci are integers and the Vi are irre- Thus, algebraic equivalence is an algebraic ana- ducible m-dimensional subvarieties of V . The log of homotopy, though the analogy is not par- cycle is said to be effective or positive if all the ticularly fruitful. coefficients ci are non-negative. The support of Algebraic equivalence has the important the cycle is the union of the subvarieties hav- property of preserving the degree of divisors; ing non-zero coefficients. The set of cycles of that is, two algebraically equivalent divisors have dimension m forms an Abelian group under ad- the same degree. It also preserves principal dition, which is denoted Zm(V ). divisors; that is, if one divisor of an algebraically equivalent pair is principal, then so is the algebraic dependence The property shared other one. (A divisor is principal if it is the di- by a set of elements in a field, when they sat- visor of a rational function.) Thus, the group isfy a non-trivial polynomial equation. Such an D0/P is a subgroup of the divisor class group 0 equation demonstrates that the set of elements Cl (X) = D/P. Here, D0 is the group of divi- is not algebraically independent. sors algebraically equivalent to 0, P is the group of principal divisors, and D is the group of di- algebraic differential equation (1) An equa- visors of degree 0. The group D0/p is exactly tion of the form the subgroup of the divisor class group realized

F x,y,y ,y ,...,y(n) = 0 by the group of points of the Picard variety of X. See algebraic family of divisors, divisor. See in which F is a polynomial with coefficients that also integral divisor, irreducible variety, Picard are complex analytic functions of x. variety. (2) An equation obtained by equating to zero a differential polynomial in a set of differential algebraic expression An expression formed variables in a differential extension field of a from the elements of a field and one or more differential field. See also differential field. variables (variables are also often called inde- terminants) using the algebraic operations of ad- algebraic element If K is an extension field dition, subtraction, multiplication, division, and of the field k, an element x ∈ K is an algebraic root extraction.

c 2001 by CRC Press LLC algebraic extension An extension field K of algebraic group An algebraic variety, to- a field k such that every α in K, but not in k, gether with group operations that are regular is algebraic over k, i.e., satisfies a polynomial functions. See regular function. equation with coefficients in k. algebraic homotopy group A generalization algebraic family A family of cycles {X(t) : of the concept of homotopy group, defined for t ∈ T } on a non-singular algebraic variety V , an algebraic variety over a field of characteris- parameterized by t ∈ T , where T is another tic p>0, formed in the context of finite étale non-singular algebraic variety, such that there coverings. is a cycle Z in V × T for which each X(t) is the projection to V of the intersection of Z and algebraic identity An algebraic equation in- ×{ } V t . volving a variable or variables that reduces to an arithmetical identity for all substitutions of algebraic family of divisors A family of di- numerical values for the variable or variables. visors ft , t ∈ T , on an irreducible variety X, where the index set T is also an irreducible va- algebraic independence Let k be a subfield = ∗ riety, and where ft φt (D) for some fixed of the field K. The elements a ,a ,...,a of K × ∈ 1 2 n divisor D on X T and all t T . Here, for are said to be algebraically independent over k ∈ ∗ each t T , φt is the map from divisors on if, for any polynomial p(X ,X ,...,X ) with × 1 2 n X T to divisors on X induced by the embed- coefficients in k, p(a ,a ,...,a ) = 0 implies : → × = 1 2 n ding φt X X T , where φ(t) (x, t), p ≡ 0. When a set of complex numbers is said × and X T is the Cartesian product of X and T . to be algebraically independent, the field k is The variety T is called the base for the algebraic understood to be the rational numbers. family ft , t ∈ T . See also Cartesian product, irreducible variety. algebraic integer A complex number that satisfies some monic polynomial equation with algebraic function A function Y = f(X , 1 integer coefficients. X2,...,XN ) satisfying an equation R(X1,X2, ...,XN ,Y)= 0 where R is a rational function over a field F . See also rational function. algebraic Lie algebra Let k be a field. An algebraic group G, realized as a closed subgroup algebraic function field Let F be a field. of the general linear group GL(n, k), is called a Any finite extension of the field of rational func- linear algebraic group, and its tangent space at tions in the identity, when given the natural Lie algebra structure, is called an algebraic Lie algebra. X1,X2,...,Xn over the field F is called an algebraic function algebraic multiplication In elementary al- field over F . gebra, the multiplication of algebraic expressions, which extends the operation of multipli- algebraic fundamental group A generaliza- cation of numbers in arithmetic. tion of the concept of fundamental group defined for an algebraic variety over a field of character- algebraic multiplicity The multiplicity of an istic p>0, formed in the context of finite étale eigenvalue λ of a matrix A as a root of the char- coverings. acteristic polynomial of A. See also geometric multiplicity, index. algebraic geometry Classically, algebraic geometry has meant the study of geometric prop- algebraic number A complex number z is erties of solutions of algebraic equations. In an algebraic number if it satisfies a non-trivial modern times, algebraic geometry has become polynomial equation P(z) = 0, for which the synonymous with the study of geometric objects coefficients of the polynomial are rational num- associated with commutative rings. bers.

c 2001 by CRC Press LLC algebraic number field A field F ⊂ C, algebraic torus An algebraic group, isomor- which is a finite degree extension of the field phic to a direct product of the multiplicative of rational numbers. group of a universal domain. A universal domain is an algebraically closed field of infinite algebraic operation In elementary algebra, transcendence degree over the prime field it con- the operations of addition, subtraction, multipli- tains. cation, division, and root extraction. In a general algebraic system A, an algebraic operation algebraic variety Classically, the term “al- may be any function from the n-fold cartesian gebraic variety” has meant either an affine al- product An to A, where n ∈{1, 2,...} (the case gebraic set or a projective algebraic set, but in n = 0 is sometimes also allowed). See also the second half of the twentieth century, various algebraic system. more general definitions have been introduced. One such more general definition, in terms of algebraic pencil A linear system of divi- sheaf theory, considers an algebraic variety V O sors in a projective variety such that one divisor to be a pair (T , ), in which T is a topological O passes through any point in general position. space and is a sheaf of germs of mappings from V into a given field k, for which the topo- { }N algebraic scheme An algebraic scheme is a logical space has a finite open cover Ui i=1 O| scheme of finite type over a field. Schemes are such that each (Ui, Ui) is isomorphic to an generalizations of varieties, and the algebraic affine variety and for which the image of V un- schemes most closely resemble the algebraic va- der the diagonal map is Zariski closed. See also rieties. See scheme. abstract algebraic variety. algebra isomorphism An algebra homomor- algebraic space A generalization of scheme phism that is also a one-to-one and onto mapping and of algebraic variety due to Artin and in- between the algebras. See algebra homomor- troduced to create a category which would be phism. closed under various constructions. Specifically, an algebraic space of finite type is an affine algebra of matrices The n×n matrices with scheme U and a closed subscheme R ⊂ U × U entries taken from a given field together with the that is an equivalence relation and for which both operations of matrix addition and matrix multi- the coordinate projections of R onto U are étale. plication. Also any nonempty set of such ma- See also étale morphism. trices, closed under those operations and containing additive inverses, and thus forming an algebraic subgroup A Zariski closed subalgebra. group of an affine algebraic group. algebra of vectors The vectors in algebraic surface A two-dimensional alge- three-dimensional space, together with the oper- braic variety. See also algebraic variety. ations of vector addition, scalar multiplication, the scalar product (also called the inner prod- algebraic system A set A, together with var- uct or the dot product), the vector product (also ious operations and relations, where by an oper- called the cross product), and the vector triple ation we mean a function from the n-fold carte- product. sian product An to A, for some n ∈{0, 1, 2,...}. algebroidal function An analytic function f algebraic system in the wider sense While satisfying a non-trivial algebraic equation an algebraic system is a set A, together with n n−1 various operations and relations on A, an alge- a0(z)f + a1(z)f +···+an(z) = 0 , braic system in the wider sense may also include higher level structures constructed by the power in which the coefficients aj (z) are meromorphic set operation. functions in a domain in the complex z-plane.

c 2001 by CRC Press LLC all-integer algorithm An algorithm for two equations which the entire calculation will be carried out = = in integers, provided the given data is all given ax1 b and x2a b in integers. Such algorithms are of interest for are uniquely solvable for x1 and x2. Also called linear programming problems that involve addi- alternative skew-field. tional integrality conditions. A notable example of such an algorithm was given in the early 1960s amalgamated product Given a family of by Gomory. groups {Gα}α∈A and embeddings {hα}α∈A of a fixed group H into the Gα, the amalgamated allowed submodule In a module M with op- product is the group G, unique up to isomor- erator domain A,anallowed submodule is a sub- phism, having the universal properties that (i.) module N ⊂ M such that a ∈ A and x ∈ N there exist homomorphisms {gα}α∈A such that implies ax ∈ N. Also called an A-submodule. gα ◦ hα = gβ ◦ hβ for all α, β ∈ A and (ii.) for any family {α}α∈A of homomorphisms of almost integral Let R be a subring of the the groups G to a fixed group L satisfying ∈ α ring R . An element a R is said to be almost ◦ h = ◦ h for all α, β ∈ A, there exists ∈ α α β β integral over R if there exists an element b R a unique homomorphism : G → L such that which is not a zero divisor and for which anb ∈ α = ◦ gα. R holds for every positive integer n. For the case of two groups G1 and G2 with isomorphic subgroups H ⊂ G and H ⊂ G , alternating group For fixed n, the subgroup 1 1 2 2 the amalgamated product of the groups can be of the group of permutations of {1, 2, ...,n}, identified with the set of finite sequences of el- consisting of the even permutations. More spe- ements of the union of the two groups with the cifically, the set of permutations σ :{1, 2,..., equivalence relation generated by identifying a n}→{1, 2,...,n} such that sequence with the sequence formed when adja- cent elements are replaced by their product if (σ (j) − σ(i))> 0 . they are in the same Gi or with the sequence 1≤i

c 2001 by CRC Press LLC (n) S ∈ CR and n a nonnegative integer, let S a function has a convergent power series expan- denote the n-fold tensor product of S over R. sion about each point of its domain. (n+1) For n a nonnegative integer, let Ei : S → (n+2) S (i = 0, 1,...,n) be the CR-morphisms analytic homomorphism A homomorphism defined by between two Lie groups which is also an analytic function (i.e., expandable in a power series E ⊗···⊗ = i (x0 xn) at each point in the Lie group, using a local co- x0 ⊗···⊗xi−1 ⊗ 1 ⊗ xi ⊗···⊗xn . ordinate system). + + Define dn : F(S(n 1)) → F(S(n 2)) by setting analytic isomorphism An analytic n homomorphism between two Lie groups which n i is one-to-one, onto and has an inverse that is d = (−1) F (Ei) . i=0 also an analytic homomorphism. See analytic homomorphism. Then {F(S(n+1)), dn} defines a cochain complex called the Amitsur complex and the coho- analytic structure A structure on a differen- mology groups are called the Amitsur cohomol- tiable manifold M which occurs when there is ogy groups. an atlas of charts {(Ui,ϕi) : i ∈ I} on M, where the transition functions Amitsur cohomology groups See Amitsur − ϕ ◦ ϕ 1 : ϕ U ∩ U → ϕ U ∩ U cohomology. j i i i j j i j are analytic. Amitsur complex See Amitsur cohomology. analytic variety A set that is the simulta- ample See ample vector bundle, ample divi- neous zero set of a finite collection of analytic sor. functions. ample divisor A divisor D such that nD is analytic vector A vector v in a Hilbert space very ample for some positive integer n. A divi- H is called an analytic vector for a finite set { }m H sor is very ample if it possesses a certain type of Tj j=1 of (unbounded) operators on if there canonical projective immersion. exist positive constants C and N such that T ···T v ≤ CNkk! ample vector bundle A vector bundle E j1 jk H ∨ where the line bundle OE∨ (1) on P(E ) is am- for all ji ∈{1,...,m} and every positive integer ple. That is, there is a morphism f from P(E∨) k. n ⊗m ∗ to a projective space P with OE∨ (1) = f O V P N (1). anisotropic A vector space with an inner product (·, ·) and containing no non-zero iso- amplification The process of increasing the tropic vector. A vector x ∈ V is isotropic if magnitude of a quantity. (x, x) = 0. analytically normal ring An analytically un- antiautomorphism An isomorphism of an ramified ring that is also integrally closed. See algebra A onto its opposite algebra A◦. See analytically unramified ring. opposite. analytically unramified ring A local ring antiendomorphism A mapping τ from a ring such that its completion contains no non-zero R to itself, which satisfies nilpotent elements. (An element x of a ring is τ(x + y) = τ(x)+ τ(y), τ(xy) = τ(y)τ(x) nilpotent if x · x = 0.) for all x,y ∈ R. The mapping τ can also be analytic function Same as a holomorphic viewed as an endomorphism (linear mapping) function, but with emphasis on the fact that such from R to its opposite ring R◦. See opposite.

c 2001 by CRC Press LLC antihomomorphism A mapping σ from a (iv.) if two apartments and contain two group G into a group H that satisfies σ(xy) = elements A, A ∈ , then there exists an iso- σ(y)σ(x) for all x,y ∈ G. An antihomor- morphism of onto which leaves invariant phism can also be viewed as a homomorphism A, A and all their faces. σ : G → H ◦ where H ◦ is the opposite group to H . See opposite. approximate functional equations Equa- tions of the form f(x) = g(x) + Ev(x) where anti-isomorphism A one-to-one, surjective f(x) and g(x) are known functions and the map f : X → Y that reverses some intrinsic growth of Ev(x) is known. property common to X and Y .IfX and Y are ∗ groups or rings, then f reverses multiplication, approximately finite algebra A C -algebra = that is the uniform closure of a finite dimensional f(ab) f(b)f(a).IfX and Y are lattices, then ∗ f reverses the lattice operations, f(a ∩ b) = C -algebra. f(a)∪ f(b)and f(a∪ b) = f(a)∩ f(b). approximately finite dimensional von Neu- mann algebra A von Neumann algebra, M, antilogarithm For a number y and a base b, = which contains an increasing sequence of finite the number x such that logb x y. dimensional subalgebras, An ⊆ An+1, such that ∪∞ A is dense in M. (Density is defined in antipode Let S be a sphere in Euclidean n=1 n terms of any of a number of equivalent topolo- space and s a point of S. The line through s and ∗ gies on M, e.g., the weak topology, or the the center of the sphere will intersect the sphere strong operator topology in any normal repre- in a uniquely determined second point s that is sentation.) called the antipode of s. The celebrated Borsuk- Ulam Theorem of algebraic topology consid- approximate number A numerical approx- →− ers the antipodal map P P . The theory imation to the actual value. of Hopf algebras contains a notion of antipode which is analogous to the geometric one just de- approximation theorem A theorem which scribed. states that one class of objects can be approximated by elements from another (usually antisymmetric decomposition The decom- smaller) class of objects. A famous example position of a compact Hausdorff space X con- is the following. sists of disjoint, closed, maximal sets of antisymmetry with respect to A, where A is a closed Weierstrass A. T. Every con- subalgebra of C(X), the algebra of all complex- tinuous function on a closed inter- valued continuous functions on X. A is called val can be uniformly approximated antisymmetric if, from the condition that f, f¯ ∈ by a polynomial. That is, if f(x) A, it follows that f is a constant function. A is continuous on the closed inter- subset S§X is called a set of antisymmetry with val [a,b] and >0, then there ex- respect to A if any function f ∈ A that is real ists a polynomial p(x) such that on S is constant on this set. |f(x) − p(x)| <for all x ∈ [a,b]. apartment An element of A, a set of sub- complexes of a complex such that the pair (, A) is a building. That is, if the following Arabic numerals The numbers 0, 1, 2, 3, 4, hold: 5, 6, 7, 8, and 9. These numbers can be used to represent all numbers in the decimal system. (i.) is thick; (ii.) the elements of A are thin chamber com- arbitrary constant A constant that can be set plexes; to any desired value. For example, in the calcu- (iii.) any two elements of belong to an lus expression 2xdx = x2 + C, the symbol apartment; C is an arbitrary constant.

c 2001 by CRC Press LLC arc cosecant The multiple-valued inverse of is satisfied. the trigonometric function csc θ, e.g., arccsc(2) (iii.) For each x ∈ F there exists a positive = π/6 + 2kπ where k is an arbitrary integer integer n such that n · 1 >x. (k = 0 specifies the principal value of arc cose- The rational numbers are an Archimedian or- cant). The principal value yields the length of dered field, and so are the real numbers. The the arc on the unit circle, subtending an angle, p-adic numbers are a non-Archimedian ordered whose cosecant equals a given value. field. The arc cosecant function is also denoted − csc 1x. Archimedian valuation A valuation on a ring R, for which v(x − y) ≤ max(v(x), v(y)) arc cosine The multiple-valued inverse of the is false, for some x,y ∈ R. See valuation. trigonometric function cos θ, e.g., arccos(−1) = π +2kπ where k is an arbitrary integer (k = 0 arcsecant The multiple-valued inverse of specifies the principal value of arc cosine). The the trigonometric function sec x, sometimes de- principal value yields the length of the arc on the noted sec−1x. unit circle, subtending an angle, whose cosine equals a given value. arc sine The multiple-valued inverse of the The arc cosine function is also denoted trigonometric function sin θ, e.g., arcsin(1) = − cos 1x. π/2+2kπ where k is an arbitrary integer (k = 0 specifies the principal value of arc sine). The arc cotangent The multiple-valued inverse principal value yields the length of the arc on of the√ trigonometric function cotan θ, e.g., arc- the unit circle, subtending an angle, whose sine cot ( 3) = π/6 + 2kπ where k is an arbitrary equals a given value. integer (k = 0 specifies the principal value of The arc sine function is also denoted sin−1x. arc cotangent). The principal value yields the length of the arc on the unit circle, subtending arc tangent The multiple-valued inverse of an angle, whose cotangent equals a given value. the√ trigonometric function tan θ, e.g., arctan The arc cotangent function is also denoted ( 3) = π/3 + 2kπ where k is an arbitrary in- − cot 1x. teger (k = 0 specifies the principal value of arc tangent). The principal value yields the length Archimedian ordered field If K is an or- of the arc on the unit circle, subtending an angle, dered field and F a subfield with the property whose tangent equals a given value. that no element of K is infinitely large over F , The arc tangent function is also denoted then we say that K is Archimedian. tan−1x.

Archimedian ordered field A set which, in ArensÐRoyden Theorem Let C(MA) denote addition to satisfying the axioms for a field, also the continuous functions on the maximal ideal possesses an Archimedian ordering. That is, the space MA of the Banach algebra A. Suppose field F is ordered in that it contains a subset P that f ∈ C(MA) and f does not vanish. Then and the following properties hold: there exists a g ∈ A, for which g−1 ∈ A, and for { } − (i.) F is the disjoint union of P , 0 , and P . which f/gˆ has a continuous logarithm on M . ∈ A In other words, each x F belongs either to P , (Here gˆ denotes the Gelfand transform of g.) or equals 0, or −x belongs to P , and these three possibilities are mutually exclusive. arithmetic The operations of addition, sub- (ii.) If x,y ∈ P , then x +y ∈ P and xy ∈ P . traction, multiplication, and division and their The ordered field is also Archimedian in that properties for the integers. the absolute value function  arithmetical equivalence An equivalence  ∈ x, if x P relation on the integers which is consistent with |x|= 0, if x = 0 the four operations of arithmetic. (a ∼ b and  −x, if x ∈−P c ∼ d imply a ± c ∼ b ± d, etc.) An example

c 2001 by CRC Press LLC would be congruence mod n where n is a positive arithmetic subgroup For a real algebraic integer. Here, two integers j and k are equiva- group G ⊂ GL(n, R), a subgroup of G, com- lent if j − k is divisible by n. See equivalence mensurable with GZ = G ∩ GL(n, R). That relation. is, : ∩ ∞ : ∩ ∞ arithmetically effective Referring to a divi- [ GZ] < and [GZ GZ] < . sor on a nonsingular algebraic surface, which is numerically semipositive, or numerically effec- Arrow-Hurewicz-Uzawa gradient method tive (nef). A technique used in solving convex or concave programming problems. Suppose ψ(x,u) is arithmetic crystal class For an n-dimen- concave or convex in x ∈ A ⊂ Rn and convex in sional Euclidean space V , an equivalence class u ∈ 0 ⊂ Rm. Usually ϕ(x,u) = ψ(x)+u·g(x) of pairs (, G) where is a lattice in V and where ϕ is the function we wish to minimize or G is a finite subgroup of O(V). Two pairs maximize and our constraints are given by the (1,G1) and (2,G2) are equivalent if there functions g (x) ≤ 01≤ j ≤ m. The method ∈ = j is a g GL(V ) such that g1 2, and devised by Arrow-Hurewicz and Uzawa consists −1 = gG1g G2. of solving the system of equations    =  arithmetic genus An integer, defined in terms  0ifxi 0   ∂ψ  of the characteristic polynomial of a homoge- dxi and ∂x < 0, U = i neous ideal in the ring of polynomials, dt  i = 1,...,n  [ ]   k x1,...,xn , in the variables x1,...,xn over ∂ψ otherwise a commutative ring k.Ifχ(¯ U; q) denotes this ∂xi characteristic polynomial, then    0ifuj = 0   ∂ψ  ¯ U; = q + q +···+ q + duj and ∂u > 0, χ( q) a0 r a1 r−1 ar−1 1 ar = j dt  j = 1,...,m   −  ∈ { q } ∂ψ otherwise where a0,...,ar k and (j ) are the binomial ∂uj − r − coefficients. The integer ( 1) (ar 1) is the If (x(t), u(t)) is a solution of this system, un- U arithmetic genus of . der certain conditions, lim x(t) = x solves the t→∞ arithmetic mean For a positive integer n, the programming problem. arithmetic mean of the n real numbers a ,..., 1 artificial variable A variable that is intro- a is (a +···+a )/n. n 1 n duced into a linear programming problem, in order to transform a constraint that is an inequal- arithmetic of associative algebras An area ity into an equality. For example, the problem of mathematics devoted to the study of simple of minimizing algebras over local fields, number fields, or function fields. C = 3x1 + 2x2 arithmetic progression A sequence {sn} of subject to the constraints real numbers such that 4x1 − 5x2 ≤ 7 x1 + x2 = 9 sn = sn−1 + r, for n>1 . with x1 ≥ 0, x2 ≥ 0, is transformed into The number s1 is the initial term, the number = + + r is the difference term. The general term sn C 3x1 2x2 0A1 satisfies s = s + (n − 1)r. n 1 subject to the constraints

arithmetic series A series of the form 4x1 − 5x2 + A1 = 7 ∞ ≥ = + n=1 an where for all n 1,an+1 an d. x1 + x2 + 0A1 = 9

c 2001 by CRC Press LLC p with x1 ≥ 0, x2 ≥ 0, A1 ≥ 0, by introducing K,Pai is a root of x − x − ai = 0, L/K is the artificial variable A1. This latter version is Galois, and the Galois group is an Abelian group in the standard form for a linear programming of exponent p. problem. Artin’s conjecture A conjecture of E. Artin Artin-Hasse function For k a p-adic num- that the Artin L-function L(s, ϕ) is entire in s, ber field with k0 a maximal subfield of k unram- whenever ϕ is irreducible and s = 1. See Artin ified over Qp, a an arbitrary integer in k0 and L-function. ∈ = − x k, the function E(a,x) exp L(a, x) ∞ σ i i pi where L(a, x) = = ((a ) /p )x and σ is Artin’s general law of reciprocity If K/k i 0 F the Frobenius automorphism of ko/Qp. is an Abelian field extension with conductor and AF is the group of ideals prime to the con- K/k Artinian module A (left) module for which ductor, then the Artin map A → A is a every descending sequence of (left) submodules homomorphism AF → Gal(K/k). The reciprocity law states that this homomorphism is ⊃ ⊃···⊃ ⊃ ⊃ M1 M2 Mn Mn+1 ... an isomorphism precisely when A lies in the subgroup HF of AF consisting of those ideals is finite, i.e., there exists an N such that M = n whose prime divisors split completely. That is, Mn+1 for all n ≥ N. ∼ AF /HF = Gal(K/k). Artinian ring A ring for which every de- Artin’s symbol The symbol K/k defined scending sequence of left ideals ℘ as follows. Let K be a finite Abelian Galois extension of a number field k with σ the princi- I1 ⊃ I2 ⊃···⊃In ⊃ In+1 ⊃ ... pal order of k and D the principal order of K. = K/k is finite. That is, there exists an N such that For each prime ℘ of K there is a σ ℘ In = In+1 for all n ≥ N. ∈ G =Gal(K/k) such that Aσ ≡ AN(℘) (mod ℘), A ∈ D ; Artin L-function The function L(s, ϕ), defined as follows. Let K be a finite Galois exten- K/k is called the Artin symbol of ℘ for the = ℘ sion of a number field k with G Gal(K/k). Abelian extension K/k. For an ideal a = ℘e Let ϕ : G → GL(V ) be a finite dimensional rep- of k relatively prime to the relative discriminant resentation (characteristic 0). For each prime ℘ e K/k K/k = K/k = − −s −1 of , define a ℘ . of k, set L℘(s, ϕ) det(I ϕ℘N(℘) ) , = 1 where ϕ℘ e τ∈T ϕ(σ τ), T is the inertia ascending central series A sequence of sub- | |= group of ℘, T e and σ is the Frobenius groups automorphism of ℘. Then {1}=H0

phic and has no zeros in 1. Furthermore, G −→ G that maps every element of G to the Riemann proved that ζ(s) can be extended to identity in G. A zero homomorphism is also a meromorphic function in the whole complex called the trivial homomorphism. plane whose only pole is a simple pole a s = 1. See also Riemann hypothesis. zero matrix A matrix in which every entry (2) Any of several special functions called ζ - is zero. functions that (i.) are meromorphic on the whole
c 2001 by CRC Press LLC complex plane, (ii.) have Dirichlet series ex- Denote by C(G; K) the set of all continu- pansion, (iii.) have Euler product expansion, ous complex-valued functions f on G that are and (iv.) satisfy certain functional equations. See invariant under every left translation by ele- also L-function. ments in K,byC(G,K) the set of elements in C(G; K) that are two-sided K-invariant, and zeta function defined by Hecke operators by L the subset of C(G,K) consisting of all Let Tn (n = 1, 2, 3,...) be a Hecke operator functions with compact support. Then L is an defined with respect to a discontinuous group algebra over C if the product of two elements is 0. Let M be the representation space of the defined by convolution. If λ is an algebra homo- Hecke operator ring K and denote by (Tn) the morphism from L into C, and the eigenspace matrix of the operation of Tn ∈ K on M. The {g ∈ C : f ∗g = λ(f )g for all f ∈ L} contains −s matrix-valued function n(Tn)n is called the a nonzero element, then it contains a two-sided ζ-function defined by Hecke operators. K-invariant element ω normalized by w(e) = 1. This element ω is unique and is called the zonal ZG-lattice Let G be a finite group. A finitely spherical function associated with λ. generated ZG-module which is Z-torsion free is called a ZG-lattice. Z-order Let Z be the ring of integers. Let A be a finite dimensional algebra with unit element zonal spherical function (1) A spheri- e over a field F . A subring G of A is called cal function of degree n (a harmonic polyno- a Z-order in A if it satisfies (i.) e ∈ G, (ii.) mial in three variables of degree n restricted G contains a basis of A as a vector space over to the unit sphere S2 of Euclidean space E3), the field F (an F -basis of A), and (iii.) G is a (n) 2 Zu (x1,x2,x3) (u ∈ S ), which has a constant finitely generated Z module (a finitely generated value on any circle in S2 whose plane is perpen- Abelian group). dicular to the vector u. This generalizes to the case where Z is re- (2) Let G be a locally compact unimodu- placed by an arbitrary ring R, in which case G lar group and K a compact subgroup of G. is called an R-order. See also R-basis, R-order.
c 2001 by CRC Press LLC