P-Adic Numbers and Solving P-Adic Equations
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OSTROWSKI for NUMBER FIELDS 1. Introduction in 1916, Ostrowski [6] Classified the Nontrivial Absolute Values on Q: up to Equival
OSTROWSKI FOR NUMBER FIELDS KEITH CONRAD 1. Introduction In 1916, Ostrowski [6] classified the nontrivial absolute values on Q: up to equivalence, they are the usual (archimedean) absolute value and the p-adic absolute values for different primes p, with none of these being equivalent to each other. We will see how this theorem extends to a number field K, giving a list of all the nontrivial absolute values on K up to equivalence: for each nonzero prime ideal p in OK there is a p-adic absolute value, real embeddings of K and complex embeddings of K up to conjugation lead to archimedean absolute values on K, and every nontrivial absolute value on K is equivalent to a p-adic, real, or complex absolute value. 2. Defining nontrivial absolute values on K For each nonzero prime ideal p in OK , a p-adic absolute value on K is defined in terms × of a p-adic valuation ordp that is first defined on OK − f0g and extended to K by taking ratios. m Definition 2.1. For x 2 OK − f0g, define ordp(x) := m where xOK = p a with m ≥ 0 and p - a. We have ordp(xy) = ordp(x)+ordp(y) for nonzero x and y in OK by unique factorization of × ideals in OK . This lets us extend ordp to K by using ratios of nonzero numbers in OK : for × α 2 K , write α = x=y for nonzero x and y in OK and set ordp(α) := ordp(x) − ordp(y). To 0 0 0 0 0 0 see this is well-defined, if x=y = x =y for nonzero x; y; x , and y in OK then xy = x y 0 0 in OK , which implies ordp(x) + ordp(y ) = ordp(x ) + ordp(y), so ordp(x) − ordp(y) = 0 0 × × ordp(x ) − ordp(y ). -
The Structure Theory of Complete Local Rings
The structure theory of complete local rings Introduction In the study of commutative Noetherian rings, localization at a prime followed by com- pletion at the resulting maximal ideal is a way of life. Many problems, even some that seem \global," can be attacked by first reducing to the local case and then to the complete case. Complete local rings turn out to have extremely good behavior in many respects. A key ingredient in this type of reduction is that when R is local, Rb is local and faithfully flat over R. We shall study the structure of complete local rings. A complete local ring that contains a field always contains a field that maps onto its residue class field: thus, if (R; m; K) contains a field, it contains a field K0 such that the composite map K0 ⊆ R R=m = K is an isomorphism. Then R = K0 ⊕K0 m, and we may identify K with K0. Such a field K0 is called a coefficient field for R. The choice of a coefficient field K0 is not unique in general, although in positive prime characteristic p it is unique if K is perfect, which is a bit surprising. The existence of a coefficient field is a rather hard theorem. Once it is known, one can show that every complete local ring that contains a field is a homomorphic image of a formal power series ring over a field. It is also a module-finite extension of a formal power series ring over a field. This situation is analogous to what is true for finitely generated algebras over a field, where one can make the same statements using polynomial rings instead of formal power series rings. -
Math 412. Quotient Rings of Polynomial Rings Over a Field
(c)Karen E. Smith 2018 UM Math Dept licensed under a Creative Commons By-NC-SA 4.0 International License. Math 412. Quotient Rings of Polynomial Rings over a Field. For this worksheet, F always denotes a fixed field, such as Q; R; C; or Zp for p prime. DEFINITION. Fix a polynomial f(x) 2 F[x]. Define two polynomials g; h 2 F[x] to be congruent modulo f if fj(g − h). We write [g]f for the set fg + kf j k 2 F[x]g of all polynomials congruent to g modulo f. We write F[x]=(f) for the set of all congruence classes of F[x] modulo f. The set F[x]=(f) has a natural ring structure called the Quotient Ring of F[x] by the ideal (f). CAUTION: The elements of F[x]=(f) are sets so the addition and multiplication, while induced by those in F[x], is a bit subtle. A. WARM UP. Fix a polynomial f(x) 2 F[x] of degree d > 0. (1) Discuss/review what it means that congruence modulo f is an equivalence relation on F[x]. (2) Discuss/review with your group why every congruence class [g]f contains a unique polynomial of degree less than d. What is the analogous fact for congruence classes in Z modulo n? 2 (3) Show that there are exactly four distinct congruence classes for Z2[x] modulo x + x. List them out, in set-builder notation. For each of the four, write it in the form [g]x2+x for two different g. -
Quotient-Rings.Pdf
4-21-2018 Quotient Rings Let R be a ring, and let I be a (two-sided) ideal. Considering just the operation of addition, R is a group and I is a subgroup. In fact, since R is an abelian group under addition, I is a normal subgroup, and R the quotient group is defined. Addition of cosets is defined by adding coset representatives: I (a + I)+(b + I)=(a + b)+ I. The zero coset is 0+ I = I, and the additive inverse of a coset is given by −(a + I)=(−a)+ I. R However, R also comes with a multiplication, and it’s natural to ask whether you can turn into a I ring by multiplying coset representatives: (a + I) · (b + I)= ab + I. I need to check that that this operation is well-defined, and that the ring axioms are satisfied. In fact, everything works, and you’ll see in the proof that it depends on the fact that I is an ideal. Specifically, it depends on the fact that I is closed under multiplication by elements of R. R By the way, I’ll sometimes write “ ” and sometimes “R/I”; they mean the same thing. I Theorem. If I is a two-sided ideal in a ring R, then R/I has the structure of a ring under coset addition and multiplication. Proof. Suppose that I is a two-sided ideal in R. Let r, s ∈ I. Coset addition is well-defined, because R is an abelian group and I a normal subgroup under addition. I proved that coset addition was well-defined when I constructed quotient groups. -
1 Absolute Values and Valuations
Algebra in positive Characteristic Fr¨uhjahrssemester 08 Lecturer: Prof. R. Pink Organizer: P. Hubschmid Absolute values, valuations and completion F. Crivelli ([email protected]) April 21, 2008 Introduction During this talk I’ll introduce the basic definitions and some results about valuations, absolute values of fields and completions. These notions will give two basic examples: the rational function field Fq(T ) and the field Qp of the p-adic numbers. 1 Absolute values and valuations All along this section, K denote a field. 1.1 Absolute values We begin with a well-known Definition 1. An absolute value of K is a function | | : K → R satisfying these properties, ∀ x, y ∈ K: 1. |x| = 0 ⇔ x = 0, 2. |x| ≥ 0, 3. |xy| = |x| |y|, 4. |x + y| ≤ |x| + |y|. (Triangle inequality) Note that if we set |x| = 1 for all 0 6= x ∈ K and |0| = 0, we have an absolute value on K, called the trivial absolute value. From now on, when we speak about an absolute value | |, we assume that | | is non-trivial. Moreover, if we define d : K × K → R by d(x, y) = |x − y|, x, y ∈ K, d is a metric on K and we have a topological structure on K. We have also some basic properties that we can deduce directly from the definition of an absolute value. 1 Lemma 1. Let | | be an absolute value on K. We have 1. |1| = 1, 2. |ζ| = 1, for all ζ ∈ K with ζd = 1 for some 0 6= d ∈ N (ζ is a root of unity), 3. -
4.3 Absolute Value Equations 373
Section 4.3 Absolute Value Equations 373 4.3 Absolute Value Equations In the previous section, we defined −x, if x < 0. |x| = (1) x, if x ≥ 0, and we saw that the graph of the absolute value function defined by f(x) = |x| has the “V-shape” shown in Figure 1. y 10 x 10 Figure 1. The graph of the absolute value function f(x) = |x|. It is important to note that the equation of the left-hand branch of the “V” is y = −x. Typical points on this branch are (−1, 1), (−2, 2), (−3, 3), etc. It is equally important to note that the right-hand branch of the “V” has equation y = x. Typical points on this branch are (1, 1), (2, 2), (3, 3), etc. Solving |x| = a We will now discuss the solutions of the equation |x| = a. There are three distinct cases to discuss, each of which depends upon the value and sign of the number a. • Case I: a < 0 If a < 0, then the graph of y = a is a horizontal line that lies strictly below the x-axis, as shown in Figure 2(a). In this case, the equation |x| = a has no solutions because the graphs of y = a and y = |x| do not intersect. 1 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ Version: Fall 2007 374 Chapter 4 Absolute Value Functions • Case II: a = 0 If a = 0, then the graph of y = 0 is a horizontal line that coincides with the x-axis, as shown in Figure 2(b). -
Absolute Values and Discrete Valuations
18.785 Number theory I Fall 2015 Lecture #1 09/10/2015 1 Absolute values and discrete valuations 1.1 Introduction At its core, number theory is the study of the ring Z and its fraction field Q. Many questions about Z and Q naturally lead one to consider finite extensions K=Q, known as number fields, and their rings of integers OK , which are finitely generated extensions of Z (the integral closure of Z in K, as we shall see). There is an analogous relationship between the ring Fq[t] of polynomials over a finite field Fq, its fraction field Fq(t), and the finite extensions of Fq(t), which are known as (global) function fields; these fields play an important role in arithmetic geometry, as they are are categorically equivalent to nice (smooth, projective, geometrically integral) curves over finite fields. Number fields and function fields are collectively known as global fields. Associated to each global field k is an infinite collection of local fields corresponding to the completions of k with respect to its absolute values; for the field of rational numbers Q, these are the field of real numbers R and the p-adic fields Qp (as you will prove on Problem Set 1). The ring Z is a principal ideal domain (PID), as is Fq[t]. These rings have dimension one, which means that every nonzero prime ideal is maximal; thus each nonzero prime ideal has an associated residue field, and for both Z and Fq[t] these residue fields are finite. In the case of Z we have residue fields Z=pZ ' Fp for each prime p, and for Fq[t] we have residue fields Fqd associated to each irreducible polynomial of degree d. -
Subrings, Ideals and Quotient Rings the First Definition Should Not Be
Section 17: Subrings, Ideals and Quotient Rings The first definition should not be unexpected: Def: A nonempty subset S of a ring R is a subring of R if S is closed under addition, negatives (so it's an additive subgroup) and multiplication; in other words, S inherits operations from R that make it a ring in its own right. Naturally, Z is a subring of Q, which is a subring of R, which is a subring of C. Also, R is a subring of R[x], which is a subring of R[x; y], etc. The additive subgroups nZ of Z are subrings of Z | the product of two multiples of n is another multiple of n. For the same reason, the subgroups hdi in the various Zn's are subrings. Most of these latter subrings do not have unities. But we know that, in the ring Z24, 9 is an idempotent, as is 16 = 1 − 9. In the subrings h9i and h16i of Z24, 9 and 16 respectively are unities. The subgroup h1=2i of Q is not a subring of Q because it is not closed under multiplication: (1=2)2 = 1=4 is not an integer multiple of 1=2. Of course the immediate next question is which subrings can be used to form factor rings, as normal subgroups allowed us to form factor groups. Because a ring is commutative as an additive group, normality is not a problem; so we are really asking when does multiplication of additive cosets S + a, done by multiplying their representatives, (S + a)(S + b) = S + (ab), make sense? Again, it's a question of whether this operation is well-defined: If S + a = S + c and S + b = S + d, what must be true about S so that we can be sure S + (ab) = S + (cd)? Using the definition of cosets: If a − c; b − d 2 S, what must be true about S to assure that ab − cd 2 S? We need to have the following element always end up in S: ab − cd = ab − ad + ad − cd = a(b − d) + (a − c)d : Because b − d and a − c can be any elements of S (and either one may be 0), and a; d can be any elements of R, the property required to assure that this element is in S, and hence that this multiplication of cosets is well-defined, is that, for all s in S and r in R, sr and rs are also in S. -
An Introduction to the P-Adic Numbers Charles I
Union College Union | Digital Works Honors Theses Student Work 6-2011 An Introduction to the p-adic Numbers Charles I. Harrington Union College - Schenectady, NY Follow this and additional works at: https://digitalworks.union.edu/theses Part of the Logic and Foundations of Mathematics Commons Recommended Citation Harrington, Charles I., "An Introduction to the p-adic Numbers" (2011). Honors Theses. 992. https://digitalworks.union.edu/theses/992 This Open Access is brought to you for free and open access by the Student Work at Union | Digital Works. It has been accepted for inclusion in Honors Theses by an authorized administrator of Union | Digital Works. For more information, please contact [email protected]. AN INTRODUCTION TO THE p-adic NUMBERS By Charles Irving Harrington ********* Submitted in partial fulllment of the requirements for Honors in the Department of Mathematics UNION COLLEGE June, 2011 i Abstract HARRINGTON, CHARLES An Introduction to the p-adic Numbers. Department of Mathematics, June 2011. ADVISOR: DR. KARL ZIMMERMANN One way to construct the real numbers involves creating equivalence classes of Cauchy sequences of rational numbers with respect to the usual absolute value. But, with a dierent absolute value we construct a completely dierent set of numbers called the p-adic numbers, and denoted Qp. First, we take p an intuitive approach to discussing Qp by building the p-adic version of 7. Then, we take a more rigorous approach and introduce this unusual p-adic absolute value, j jp, on the rationals to the lay the foundations for rigor in Qp. Before starting the construction of Qp, we arrive at the surprising result that all triangles are isosceles under j jp. -
Twenty-Seven Element Albertian Semifields Thomas Joel Hughes University of Texas at El Paso, [email protected]
University of Texas at El Paso DigitalCommons@UTEP Open Access Theses & Dissertations 2016-01-01 Twenty-Seven Element Albertian Semifields Thomas Joel Hughes University of Texas at El Paso, [email protected] Follow this and additional works at: https://digitalcommons.utep.edu/open_etd Part of the Mathematics Commons Recommended Citation Hughes, Thomas Joel, "Twenty-Seven Element Albertian Semifields" (2016). Open Access Theses & Dissertations. 667. https://digitalcommons.utep.edu/open_etd/667 This is brought to you for free and open access by DigitalCommons@UTEP. It has been accepted for inclusion in Open Access Theses & Dissertations by an authorized administrator of DigitalCommons@UTEP. For more information, please contact [email protected]. Twenty-Seven Element Albertian Semifields THOMAS HUGHES Master's Program in Mathematical Sciences APPROVED: Piotr Wojciechowski, Ph.D., Chair Emil Schwab, Ph.D. Vladik Kreinovich, Ph.D. Charles Ambler, Ph.D. Dean of the Graduate School Twenty-Seven Element Albertian Semifields by THOMAS HUGHES THESIS Presented to the Faculty of the Graduate School of The University of Texas at El Paso in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE Master's Program in Mathematical Sciences THE UNIVERSITY OF TEXAS AT EL PASO December 2016 Acknowledgements First and foremost, I'd like to acknowledge and thank my advisor Dr. Piotr Wojciechowski, who inspired me to pursue serious studies in mathematics and encouraged me to take on this topic. I'd like also to acknowledge and thank Matthew Wojciechowski, from the Department of Computer Science at Ohio State University, whose help and collaboration on developing a program and interface, which found semifields and their respective 2-periodic bases, made it possible to obtain some of the crucial results in this thesis. -
MAT 240 - Algebra I Fields Definition
MAT 240 - Algebra I Fields Definition. A field is a set F , containing at least two elements, on which two operations + and · (called addition and multiplication, respectively) are defined so that for each pair of elements x, y in F there are unique elements x + y and x · y (often written xy) in F for which the following conditions hold for all elements x, y, z in F : (i) x + y = y + x (commutativity of addition) (ii) (x + y) + z = x + (y + z) (associativity of addition) (iii) There is an element 0 ∈ F , called zero, such that x+0 = x. (existence of an additive identity) (iv) For each x, there is an element −x ∈ F such that x+(−x) = 0. (existence of additive inverses) (v) xy = yx (commutativity of multiplication) (vi) (x · y) · z = x · (y · z) (associativity of multiplication) (vii) (x + y) · z = x · z + y · z and x · (y + z) = x · y + x · z (distributivity) (viii) There is an element 1 ∈ F , such that 1 6= 0 and x·1 = x. (existence of a multiplicative identity) (ix) If x 6= 0, then there is an element x−1 ∈ F such that x · x−1 = 1. (existence of multiplicative inverses) Remark: The axioms (F1)–(F-5) listed in the appendix to Friedberg, Insel and Spence are the same as those above, but are listed in a different way. Axiom (F1) is (i) and (v), (F2) is (ii) and (vi), (F3) is (iii) and (vii), (F4) is (iv) and (ix), and (F5) is (vii). Proposition. Let F be a field. -
Complete Set of Algebra Exams
TIER ONE ALGEBRA EXAM 4 18 (1) Consider the matrix A = . −3 11 ✓− ◆ 1 (a) Find an invertible matrix P such that P − AP is a diagonal matrix. (b) Using the previous part of this problem, find a formula for An where An is the result of multiplying A by itself n times. (c) Consider the sequences of numbers a = 1, b = 0, a = 4a + 18b , b = 3a + 11b . 0 0 n+1 − n n n+1 − n n Use the previous parts of this problem to compute closed formulae for the numbers an and bn. (2) Let P2 be the vector space of polynomials with real coefficients and having degree less than or equal to 2. Define D : P2 P2 by D(f) = f 0, that is, D is the linear transformation given by takin!g the derivative of the poly- nomial f. (You needn’t verify that D is a linear transformation.) (3) Give an example of each of the following. (No justification required.) (a) A group G, a normal subgroup H of G, and a normal subgroup K of H such that K is not normal in G. (b) A non-trivial perfect group. (Recall that a group is perfect if it has no non-trivial abelian quotient groups.) (c) A field which is a three dimensional vector space over the field of rational numbers, Q. (d) A group with the property that the subset of elements of finite order is not a subgroup. (e) A prime ideal of Z Z which is not maximal. ⇥ (4) Show that any field with four elements is isomorphic to F2[t] .