# Design of an IDM-Based Determinant Computing Unit for a 130Nm Low Power CMOS ASIC Acoustic Localization Processor

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• Algebraic Number Theory
Algebraic Number Theory William B. Hart Warwick Mathematics Institute Abstract. We give a short introduction to algebraic number theory. Algebraic number theory is the study of extension ﬁelds Q(α1; α2; : : : ; αn) of the rational numbers, known as algebraic number ﬁelds (sometimes number ﬁelds for short), in which each of the adjoined complex numbers αi is algebraic, i.e. the root of a polynomial with rational coeﬃcients. Throughout this set of notes we use the notation Z[α1; α2; : : : ; αn] to denote the ring generated by the values αi. It is the smallest ring containing the integers Z and each of the αi. It can be described as the ring of all polynomial expressions in the αi with integer coeﬃcients, i.e. the ring of all expressions built up from elements of Z and the complex numbers αi by ﬁnitely many applications of the arithmetic operations of addition and multiplication. The notation Q(α1; α2; : : : ; αn) denotes the ﬁeld of all quotients of elements of Z[α1; α2; : : : ; αn] with nonzero denominator, i.e. the ﬁeld of rational functions in the αi, with rational coeﬃcients. It is the smallest ﬁeld containing the rational numbers Q and all of the αi. It can be thought of as the ﬁeld of all expressions built up from elements of Z and the numbers αi by ﬁnitely many applications of the arithmetic operations of addition, multiplication and division (excepting of course, divide by zero). 1 Algebraic numbers and integers A number α 2 C is called algebraic if it is the root of a monic polynomial n n−1 n−2 f(x) = x + an−1x + an−2x + ::: + a1x + a0 = 0 with rational coeﬃcients ai.
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• Fields Besides the Real Numbers Math 130 Linear Algebra
manner, which are both commutative and asso- ciative, both have identity elements (the additive identity denoted 0 and the multiplicative identity denoted 1), addition has inverse elements (the ad- ditive inverse of x denoted −x as usual), multipli- cation has inverses of nonzero elements (the multi- Fields besides the Real Numbers 1 −1 plicative inverse of x denoted x or x ), multipli- Math 130 Linear Algebra cation distributes over addition, and 0 6= 1. D Joyce, Fall 2015 Of course, one example of a ﬁeld is the ﬁeld of Most of the time in linear algebra, our vectors real numbers R. What are some others? will have coordinates that are real numbers, that is to say, our scalar ﬁeld is R, the real numbers. Example 2 (The ﬁeld of rational numbers, Q). Another example is the ﬁeld of rational numbers. But linear algebra works over other ﬁelds, too, A rational number is the quotient of two integers like C, the complex numbers. In fact, when we a=b where the denominator is not 0. The set of discuss eigenvalues and eigenvectors, we'll need to all rational numbers is denoted Q. We're familiar do linear algebra over C. Some of the applications with the fact that the sum, diﬀerence, product, and of linear algebra such as solving linear diﬀerential quotient (when the denominator is not zero) of ra- equations require C as as well. tional numbers is another rational number, so Q Some applications in computer science use linear has all the operations it needs to be a ﬁeld, and algebra over a two-element ﬁeld Z (described be- 2 since it's part of the ﬁeld of the real numbers R, its low).
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• Ring (Mathematics) 1 Ring (Mathematics)
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• A Brief History of Ring Theory
A Brief History of Ring Theory by Kristen Pollock Abstract Algebra II, Math 442 Loyola College, Spring 2005 A Brief History of Ring Theory Kristen Pollock 2 1. Introduction In order to fully deﬁne and examine an abstract ring, this essay will follow a procedure that is unlike a typical algebra textbook. That is, rather than initially oﬀering just deﬁnitions, relevant examples will ﬁrst be supplied so that the origins of a ring and its components can be better understood. Of course, this is the path that history has taken so what better way to proceed? First, it is important to understand that the abstract ring concept emerged from not one, but two theories: commutative ring theory and noncommutative ring the- ory. These two theories originated in diﬀerent problems, were developed by diﬀerent people and ﬂourished in diﬀerent directions. Still, these theories have much in com- mon and together form the foundation of today's ring theory. Speciﬁcally, modern commutative ring theory has its roots in problems of algebraic number theory and algebraic geometry. On the other hand, noncommutative ring theory originated from an attempt to expand the complex numbers to a variety of hypercomplex number systems. 2. Noncommutative Rings We will begin with noncommutative ring theory and its main originating ex- ample: the quaternions. According to Israel Kleiner's article \The Genesis of the Abstract Ring Concept," [2]. these numbers, created by Hamilton in 1843, are of the form a + bi + cj + dk (a; b; c; d 2 R) where addition is through its components 2 2 2 and multiplication is subject to the relations i =pj = k = ijk = −1.
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• 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.
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• Define the Determinant of A, to Be Det(A) = Ad − Bc. EX
DETERMINANT 1. 2 × 2 Determinants Fix a matrix A 2 R2×2 a b A = c d Deﬁne the determinant of A, to be det(A) = ad − bc: 2 −1 EXAMPLE: det = 2(2) − (−1)(−4) = 0. −4 2 det(A) encodes important geometric and algebraic information about A. For instance, the matrix A is invertible if and only if det(A) 6= 0. To see this suppose det(A) 6= 0, in this case the matrix 1 d −b B = det A −c a is well-deﬁned. One computes, 1 ad − bc 0 BA = = I : ad − bc 0 ad − bc 2 and so A is invertible and A−1 = B. In the other direction, if A is not invertible, then the columns are linearly dependent and so either a = kb and c = kd or b = ka and d = kc for some k 2 R. Hence, det(A) = kbd − kbd = 0 or = kac − kac = 0. More geometrically, if x1 S = : 0 ≤ x1; x2 ≤ 1 x2 is the unit square and A(S) = fA~x : ~x 2 Sg is the image under multiplication by A of S, then the area of A(S), which we denote by jA(S)j is jA(S)j = j det(A)j: 2. Determinants of n × n matrices We seek a good notion for det(A) when A 2 Rn×n. It is improtant to remember that while this is of historical and theoretical importance, it is not as signiﬁcant for computational purposes. Begin by ﬁxing a matrix A 2 Rn×n of the form 2 3 a11 ··· a1n 6 .
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• Number Fields
Part II | Number Fields Based on lectures by I. Grojnowski Notes taken by Dexter Chua Lent 2016 These notes are not endorsed by the lecturers, and I have modiﬁed them (often signiﬁcantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. Part IB Groups, Rings and Modules is essential and Part II Galois Theory is desirable Deﬁnition of algebraic number ﬁelds, their integers and units. Norms, bases and discriminants. [3] Ideals, principal and prime ideals, unique factorisation. Norms of ideals. [3] Minkowski's theorem on convex bodies. Statement of Dirichlet's unit theorem. Deter- mination of units in quadratic ﬁelds. [2] Ideal classes, ﬁniteness of the class group. Calculation of class numbers using statement of the Minkowski bound. [3] Dedekind's theorem on the factorisation of primes. Application to quadratic ﬁelds. [2] Discussion of the cyclotomic ﬁeld and the Fermat equation or some other topic chosen by the lecturer. [3] 1 Contents II Number Fields Contents 0 Introduction 3 1 Number ﬁelds 4 2 Norm, trace, discriminant, numbers 10 3 Multiplicative structure of ideals 17 4 Norms of ideals 27 5 Structure of prime ideals 32 6 Minkowski bound and ﬁniteness of class group 37 7 Dirichlet's unit theorem 48 8 L-functions, Dirichlet series* 56 Index 68 2 0 Introduction II Number Fields 0 Introduction Technically, IID Galois Theory is not a prerequisite of this course. However, many results we have are analogous to what we did in Galois Theory, and we will not refrain from pointing out the correspondence.
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• RING THEORY 1. Ring Theory a Ring Is a Set a with Two Binary Operations
CHAPTER IV RING THEORY 1. Ring Theory A ring is a set A with two binary operations satisfying the rules given below. Usually one binary operation is denoted `+' and called \addition," and the other is denoted by juxtaposition and is called \multiplication." The rules required of these operations are: 1) A is an abelian group under the operation + (identity denoted 0 and inverse of x denoted x); 2) A is a monoid under the operation of multiplication (i.e., multiplication is associative and there− is a two-sided identity usually denoted 1); 3) the distributive laws (x + y)z = xy + xz x(y + z)=xy + xz hold for all x, y,andz A. Sometimes one does∈ not require that a ring have a multiplicative identity. The word ring may also be used for a system satisfying just conditions (1) and (3) (i.e., where the associative law for multiplication may fail and for which there is no multiplicative identity.) Lie rings are examples of non-associative rings without identities. Almost all interesting associative rings do have identities. If 1 = 0, then the ring consists of one element 0; otherwise 1 = 0. In many theorems, it is necessary to specify that rings under consideration are not trivial, i.e. that 1 6= 0, but often that hypothesis will not be stated explicitly. 6 If the multiplicative operation is commutative, we call the ring commutative. Commutative Algebra is the study of commutative rings and related structures. It is closely related to algebraic number theory and algebraic geometry. If A is a ring, an element x A is called a unit if it has a two-sided inverse y, i.e.
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• UNIT-REGULAR ORTHODOX SEMIGROUPS by R
UNIT-REGULAR ORTHODOX SEMIGROUPS by R. B. McFADDEN (Received 5 May, 1983) Introduction. Unit-regular rings were introduced by Ehrlich [4]. They arose in the search for conditions on a regular ring that are weaker than the ACC, DCC, or finite Goldie dimension, which with von Neumann regularity imply semisimplicity. An account of unit-regular rings, together with a good bibliography, is given by Goodearl [5]. The basic definition of unit-regularity is purely multiplicative; it is simply that for each element x of a monoid S (initially a ring R with identity) there is a unit u of S for which x = xux. The concept of a unit-regular semigroup is a natural one; for example, the full transformation semigroup on a finite set, and the semigroup of endomorphisms of a finite-dimensional vector space, are unit-regular semigroups [1]. Unit-regularity has been studied by Chen and Hsieh [2], by Tirasupa [9], and by McAlister [6]. The connection between unit-regularity and finiteness conditions has been considered by D'Alarcao [3]. The problem of describing the structure of an arbitrary unit-regular semigroup S is difficult. It appears reasonable to attempt to provide such a description in terms of the group of units of S and the set of idempotents of S, and in this direction Blyth and McFadden did determine the structure of a narrow class of unit-regular semigroups. Calling a semigroup S uniquely unit orthodox if it is orthodox and, for each x in S, there exists a unique unit u of S for which x = xux, they proved that every such semigroup is a semidirect product of a group (the group of units of S) and a band (the band of idempotents of S).
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• Math 6370: Algebraic Number Theory
Math 6370: Algebraic Number Theory Taught by David Zywina Notes by David Mehrle [email protected] Cornell University Spring 2018 Last updated May 13, 2018. The latest version is online here. Contents 1 Introduction................................. 5 1.1 Administrivia............................. 8 2 Algebraic integers.............................. 9 2.1 Trace and Norm............................. 11 2.2 Complex embeddings ........................ 13 2.3 OK as an abelian group........................ 16 2.4 Discriminants............................. 18 2.5 Example: Cyclotomic Integers.................... 25 3 The ideal class group............................ 28 3.1 Unique Factorization......................... 29 3.2 Fractional Ideals............................ 34 3.3 Ramiﬁcation.............................. 40 3.4 Dedekind Domains.......................... 44 3.5 Discrete Valuation Rings....................... 46 3.6 Extensions of number ﬁelds. .................... 48 4 Geometry of Numbers (Minkowski Theory)............... 50 4.1 Algebraic integers as a lattice .................... 53 4.2 Minkowski’s Theorem........................ 54 4.3 The class group is ﬁnite........................ 57 4.4 Bounds on the Discriminant...................... 61 4.5 Bounds on the Discriminant..................... 66 5 The units of OK ............................... 68 5.1 Dirichlet Unit Theorem and Examples............... 69 5.2 Pell’s Equation.............................. 71 5.3 Proof of the Dirichlet Unit Theorem ................ 73 6 Galois
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• THE STRUCTURE of UNIT GROUPS Contents 1. Units of Z[ √ D] 1 2
THE STRUCTURE OF UNIT GROUPS DANIEL KLINE Abstract. Since the time of ancient Greece and India, Pell's equation, a2 − 2 db = 1, has been studied and its solutions sought after. Thesep solutions are signiﬁcant, because they correspond to the units of the rings Z[ d]. Along with prime numbers, units are vital in understanding the structure and especially the factorization within diﬀerent rings. Dirichlect's Unit Theorem establishes the structure of the units of a number ﬁeld as a ﬁnite abelian group. In addition to describing the nature of units, Dirichlect's Unit Theorem represents the interconnectedness of geometry (namely Minkowski's Theorem), algebra, and number theory. Contents p 1. Units of Z[ d] 1 2. Minkowski's Theorem in the Plane 3 3. Minkowski's Theorem in General 5 4. Basic Properties of Number Fields 6 5. Algebraic Integers 9 6. Real and Imaginary Embeddings 10 7. Dirichlet's Unit Theorem 10 Acknowledgments 14 References 14 p 1. Units of Z[ d] Deﬁnition 1.1. Given a ring R, u 2 R is a unit if there exists v 2 R such that u · v = 1. We denote the set of units of a ring R as R∗. Example 1.2. Consider the ring Z. It is easy to check that Z∗ = {±1g. Units are not generally as obvious as inp the case of Z. Throughout this section, we will considerp the ringsp of the form Z[ d] where d is a square-free integer. We deﬁne Z[ d] := fa + b d j a; b 2 Zg. For identifying units of rings in this form, we will rely on the useful property that the norm of a unit, which we will deﬁne below, is equal to ±1.
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• POLYNOMIALS DEFINING MANY UNITS Let G Be a Group and Let ZG
POLYNOMIALS DEFINING MANY UNITS OSNEL BROCHE AND ANGEL´ DEL R´IO Abstract. We classify the polynomials with integral coeﬃcients that, when evaluated on a group element of ﬁnite order n, deﬁne a unit in the integral group ring for inﬁnitely many positive integers n. We show that this happens if and only if the polynomial deﬁnes generic units in the sense of Marciniak and Sehgal. We also classify the polynomials with integral coeﬃcients which provides units when evaluated on n-roots of a ﬁxed integer a for inﬁnitely many positive integers n. Let G be a group and let ZG denote the integral group ring of G. If x 2 G and f is a polynomial in one variable with integral coeﬃcients then f(x) denotes the element of ZG obtained when f is evaluated in x. This paper deals with the problem of when f(x) is a unit of ZG. It is easy to see that this only depends on the order (possibly inﬁnite) of x. If the order of x is inﬁnite and f(x) is a unit then necessarily f = ±Xm for some non-negative integer m. Hence we are only interested in the case where x has ﬁnite order. In case f(x) is a unit of ZG and n is the order of x then we say that f deﬁnes a unit on order n. Marciniak and Sehgal introduced the following deﬁnition [MS05]. A polynomial f deﬁnes generic units if there is a positive integer D such that f deﬁnes a unit on every order coprime with D.
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