Dictionary of algebra, arithmetic, and trigonometry

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 PUBLISHED VOLUMES Analysis, Calculus, and Differential Equations Douglas N. Clark Algebra, Arithmetic and Trigonometry Steven G. Krantz FORTHCOMING VOLUMES Classical & Theoretical Mathematics Catherine Cavagnaro and Will Haight Applied Mathematics for Engineers and Scientists Emma Previato The Comprehensive Dictionary of Mathematics Douglas N. Clark 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 Boca Raton London New York Washington, D.C. CRC Press This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. 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PREFACE 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 Grand Valley State University Allendale, Michigan Gerardo Aladro Florida International University Miami, Florida Mohammad Azarian University of Evansville Evansville. Indiana Susan Barton West Virginia Institute of Technology Montgomery, West Virginia Albert Boggess Texas A&M University College Station, Texas Robert Borrelli Harvey Mudd College Claremont, California Stephen W. Brady Wichita State University Wichita, Kansas Der Chen Chang Georgetown University Washington, D.C. Stephen A. Chiappari Santa Clara University Santa Clara. California Joseph A. Cima The University of North Carolina at Chapel Hill Chapel Hill, North Carolina Courtney S. Coleman Harvey Mudd College Claremont, California John B. Conway University of Tennessee Knoxville, Tennessee Neil K. Dickson University of Glasgow Glasgow, United Kingdom David E. Dobbs University of Tennessee Knoxville, Tennessee Marcus Feldman Washington University St. Louis, Missouri Stephen Humphries Brigham Young University Provo, Utah Shanyu Ji University of Houston Houston, Texas Kenneth D. Johnson University of Georgia Athens, Georgia Bao Qin Li Florida International University Miami, Florida Robert E. MacRae University of Colorado Boulder, Colorado Charles N. Moore Kansas State University Manhattan, Kansas Hossein Movahedi-Lankarani Pennsylvania State University Altoona, Pennsylvania Shashikant B. Mulay University of Tennessee Knoxville, Tennessee Judy Kenney Munshower Avila College Kansas City, Missouri c  2001 by CRC Press LLC Charles W. Neville CWN Research Berlin, Connecticut Daniel E. Otero Xavier University Cincinnati, Ohio Josef Paldus University of Waterloo Waterloo, Ontario, Canada Harold R. Parks Oregon State University Corvallis, Corvallis, Oregon Gunnar Stefansson Pennsylvania State University Altoona, Pennsylvania Anthony D. Thomas University of Wisconsin Platteville. Wisconsin Michael Tsatsomeros University of Regina Regina, Saskatchewan, Canada James S. Walker University of Wisconsin at Eau Claire Eau Claire, Wisconsin C. Eugene Wayne Boston University Boston, Massachusetts Kehe Zhu State University of New York at Albany Albany, New York c  2001 by CRC Press LLC A A-balanced mapping Let M be a right mod- ule over the ring A, and let N be a left module over the samering A. A mapping φ fromM ×N to an Abelian group G is said to be A-balanced if φ(x,·) is a group homomorphism from N to G for each x ∈ M,ifφ(·,y) is a group homo- morphism from M to G for each y ∈ N, and if φ(xa,y) = φ(x,ay) holds for all x ∈ M, y ∈ N , and a ∈ A. A-B-bimodule An Abelian group G that is a left module over the ring A and a right module over the ring B and satisfies the associative law (ax)b = a(xb) for all a ∈ A, b ∈ B, and all x ∈ G. Abeliancohomology Theusualcohomology with coefficients in an Abelian group; used if the context requires one to distinguish between the usual cohomology and the more exotic non- Abelian cohomology. See cohomology. Abeliandifferentialof thefirstkind Aholo- morphic differential on a closed Riemann sur- face; that is, a differential of the form ω = a(z)dz, where a(z) is a holomorphic function. Abelian differential of the second kind A meromorphic differential on a closed Riemann surface,thesingularitiesof which arealloforder greater than or equal to 2; that is, a differential of the form ω = a(z) dz where a(z) is a mero- morphic function with only 0 residues. Abelian differential of the third kind A differential on a closed Riemann surface that is not an Abelian differential of the first or sec- ond kind; that is, a differential of the form ω = a(z)dz where a(z) is meromorphic and has at least one non-zero residue. Abelian equation A polynomial equation f(X) = 0 is said to be an Abelian equation if its Galoisgroup isan Abelian group. See Galois group. See also Abelian group. Abelian extension A Galois extension of a field is called an Abelian extension if its Galois group is Abelian. See Galois extension. See also Abelian group. Abelian function A function f(z 1 ,z 2 ,z 3 , ,z n ) meromorphic on C n for which there ex- ist 2n vectors ω k ∈ C n , k = 1, 2, 3, ,2n, called period vectors, that are linearly indepen- dent over R and are such that f ( z + ω k ) = f(z) holds for k = 1, 2, 3, ,2n and z ∈ C n . Abelian function field The set of Abelian functions on C n corresponding to a given set of period vectors forms a field called an Abelian function field. Abeliangroup Briefly, acommutativegroup. More completely, a setG, together with abinary operation, usually denoted “+,” a unary opera- tion usually denoted “−,” and a distinguished element usually denoted “0” satisfying the fol- lowing axioms: (i.) a + (b +c) = (a +b) +c for all a, b,c ∈ G, (ii.) a + 0 = a for all a ∈ G, (iii.) a + (−a) = 0 for all a ∈ G, (iv.) a + b = b +a for all a,b ∈ G. The element 0 is called the identity, −a is called the inverse of a, axiom (i.) is called the associative axiom, and axiom (iv.) is called the commutative axiom. Abelianideal An ideal inaLiealgebrawhich forms a commutative subalgebra. Abelian integral of the first kind An indef- inite integral W(p) =  p p 0 a(z)dz on a closed Riemann surface in which the function a(z) is holomorphic (the differential ω(z) = a(z) dz is said to be an Abelian differential of the first kind). Abelian integral of the second kind An in- definiteintegralW(p) =  p p 0 a(z)dzonaclosed Riemann surface in which the function a(z) is c  2001 by CRC Press LLC meromorphic with all its singularities of order at least 2 (the differential a(z) dz is said to be an Abelian differential of the second kind). Abelian integral of the third kind An in- definiteintegralW(p) =  p p 0 a(z)dzonaclosed Riemann surface in which the function a(z) is meromorphic and has at least one non-zero resi- due(thedifferentiala(z)dz issaidto beanAbel- ian differential of the third kind). Abelian Lie group A Lie group for which the associated Lie algebra is Abelian. See also Lie algebra. Abelian projection operator A non-zero projection operator E ina vonNeumann algebra M such that the reduced von Neumann algebra M E = EME is Abelian. Abelian subvariety A subvariety of an Abelian variety that is also a subgroup. See also Abelian variety. Abelian surface A two-dimensional Abelian variety. See also Abelian variety. Abelian variety A complete algebraic vari- ety G that also forms a commutative algebraic group. That is, G is a group under group oper- ations that are regular functions. The fact that an algebraic group is complete as an algebraic variety implies that the group is commutative. See also regular function. Abel’s Theorem Niels Henrik Abel (1802- 1829) proved several results now known as “Abel’s Theorem,” but perhaps preeminent among these is Abel’s proof that the general quintic equation cannot be solved algebraically. Other theorems that may be found under the heading “Abel’s Theorem” concern power se- ries, Dirichlet series, and divisors on Riemann surfaces. absolute class field Let k be an algebraic number field. A Galois extension K of k is an absolute class field if it satisfies the following property regarding prime ideals of k: A prime ideal p of k of absolute degree 1 decomposes as the product of prime ideals of K of absolute degree 1 if and only if p is a principal ideal. The term “absolute class field” is used to dis- tinguish the Galois extensions described above, which were introduced by Hilbert, from a more general concept of “class field” defined by Tagaki. See also class field. absolute covariant A covariant of weight 0. See also covariant. absolute inequality An inequality involving variables that is valid for all possible substitu- tions of real numbers for the variables. absolute invariant Any quantity or property of an algebraic variety that is preserved under birational transformations. absolutely irreducible character The char- acter of an absolutely irreducible representation. A representation is absolutely irreducible if it is irreducible and if the representation obtained by making anextension of theground fieldremains irreducible. absolutely irreducible representation A representation is absolutely irreducible if it is irreducible and if the representation obtained by making anextension of theground fieldremains irreducible. absolutely simple group A group that con- tains no serial subgroup. The notion of an ab- solutely simple group is a strengthening of the concept of a simple group that is appropriate for infinite groups. See serial subgroup. absolutely uniserial algebra Let A be an al- gebra over the field K, and let L be an extension field of K. Then L ⊗ K A can be regarded as an algebra over L. If, for every choice of L, L ⊗ K A can be decomposed into a direct sum of ideals which are primary rings, then A is an absolutely uniserial algebra. absolute multiple covariant A multiple co- variant of weight 0. See also multiple covari- ants. c  2001 by CRC Press LLC absolute number A specific number repre- sented by numerals such as 2, 3 4 , or 5.67 in con- trast with a literal number which is a number represented by a letter. absolute value of a complex number More commonly called the modulus, the absolute val- ue of the complex number z = a + ib, where a and b are real, is denoted by |z| and equals the non-negative real number √ a 2 + b 2 . absolute value of a vector More commonly called the magnitude, the absolute value of the vector −→ v = ( v 1 ,v 2 , ,v n ) is denoted by | −→ v | and equals the non-negative real number  v 2 1 + v 2 2 +···+v 2 n . absolute value of real number For a real numberr,the nonnegativerealnumber|r|,given by |r|=  r if r ≥ 0 −r if r<0 .  abstract algebraicvariety A set that is anal- ogous to an ordinary algebraic variety, but de- fined only locally and without an imbedding. abstract function (1) In the theory of gen- eralized almost-periodic functions, a function mapping R to a Banach space other than the complex numbers. (2) A function from one Banach space to an- other Banach space that is everywhere differen- tiable in the sense of Fréchet. abstract variety A generalization of the no- tion of an algebraic variety introduced by Weil, in analogy with the definition of a differentiable manifold. An abstract variety (also called an abstract algebraic variety) consists of (i.) a family {V α } α∈A of affine algebraic sets over a given field k, (ii.) for each α ∈ A a family of open subsets {W αβ } β∈A of V α , and(iii.) for each pair α and β in A a birational transformation be- tween W αβ and W αβ such that the composition of the birational transformations between sub- sets of V α and V β and between subsets of V β and V γ are consistent with those between sub- sets of V α and V γ . accelerationparameter Aparameter chosen in applying successive over-relaxation (which is an accelerated version of the Gauss-Seidel method)tosolveasystem of linearequationsnu- merically. Morespecifically, onesolvesAx = b iteratively by setting x n+1 = x n + R ( b − Ax n ) , where R =  L + ω −1 D  −1 with L the lower triangular submatrix of A, D the diagonal of A, and 0 <ω<2. Here, ω is the acceleration parameter, also called the relaxation parameter. Analysis is required to choose an appropriate value of ω. acyclic chain complex An augmented, pos- itive chain complex ··· ∂ n+1 −→ X n ∂ n −→ X n−1 ∂ n−1 −→ ··· ∂ 2 −→ X 1 ∂ 1 −→ X 0  → A → 0 forming an exact sequence. This in turn means that the kernel of ∂ n equals the image of ∂ n+1 for n ≥ 1, the kernel of  equals the image of ∂ 1 , and  is surjective. Here the X i and A are modules over a commutative unitary ring. addend In arithmetic, a number that is to be added to another number. In general, one of the operands of an operation of addition. See also addition. addition (1) A basic arithmetic operation that expresses the relationship between the number of elements in each of two disjoint sets and the numberof elementsin the union ofthose two sets. (2) The name of the binary operation in an Abelian group, when the notation “+” is used for that operation. See also Abelian group. (3) The name of the binary operation in a ring, under which the elements form an Abelian group. See also Abelian group. (4) Sometimes, thename of one of the opera- tions in a multi-operator group, even though the operation is not commutative. c  2001 by CRC Press LLC [...]... Cartan’s criterion of solvability Let gl(n, K) be the general linear Lie algebra of degree n over a field K and let L be a subalgebra of gl(n, K) Then L is solvable if and only if tr(AB) = 0 (tr(AB) = trace of AB), for every A ∈ L and B ∈ [L, L] Cartesian product If X and Y are sets, then the Cartesian product of X and Y , denoted X × Y , is the set of all ordered pairs (x, y) with x ∈ X and y ∈ Y Cartan’s... (1) Center of symmetry in Euclidean geometry The midpoint of a line, center of a triangle, circle, ellipse, regular polygon, sphere, ellipsoid, etc (2) Center of a group, ring, or Lie algebra X The set of all elements of X that commute with every element of X (3) Center of a lattice L The set of all central elements of L central extension Let G, H , and K be groups such that G is an extension of K by... β ∈ A and (ii.) for any family { α }α∈A of homomorphisms of the groups Gα to a fixed group L satisfying α ◦ hα = β ◦ hβ for all α, β ∈ A, there exists a unique homomorphism : G → L such that α = ◦ gα For the case of two groups G1 and G2 with isomorphic subgroups H1 ⊂ G1 and H2 ⊂ G2 , the amalgamated product of the groups can be identified with the set of finite sequences of elements of the union of the... rank of the group of rational points of an elliptic curve E is equal to the order of the 0 of L(s, E) at s = 1 Consider the elliptic curve E : y 2 = x 3 − ax − b where a and b are integers If E(Q) = E ∩ (Q × Q), by Mordell’s Theorem E(Q) is a finitely generated Abelian group Let N be the conductor of E, and if p | N, let ap + p be the number of solutions of y 2 = (x 3 − ax − b) (mod p) The L-function of. .. a group of permutations of the roots of the equation The affect of the equation is the index of the Galois group in the group of all permutations of the roots of the equation c 2001 by CRC Press LLC affectless equation A polynomial equation for which the Galois group consists of all permutations See also affect affine algebraic group group See linear algebraic affine morphism of schemes Let X and Y be... Thus, the group D0 /P is a subgroup of the divisor class group Cl 0 (X) = D/P Here, D0 is the group of divisors algebraically equivalent to 0, P is the group of principal divisors, and D is the group of divisors of degree 0 The group D0 /p is exactly the subgroup of the divisor class group realized by the group of points of the Picard variety of X See algebraic family of divisors, divisor See also integral... to V of the intersection of Z and V × {t}, and X1 = X(t1 ), X2 = X(t2 ), for some t1 , t2 ∈ T Such a family of cycles X(t) is called an algebraic family algebraic equivalence of divisors Two divisors f and g on an irreducible variety X are algebraically equivalent if there exists an algebraic family of divisors, ft , t ∈ T , and points t1 and t2 ∈ T , such that f = ft1 , and g = ft2 Thus, algebraic... is holomorphically induced from a one-dimensional holomorphic representation of a Borel subgroup of Gc boundary (1) (Topology.) The intersection of the complements of the interior and exterior of a set is called the boundary of the set Or, equivalently, a set’s boundary is the intersection of its closure and the closure of its complement (2) (Algebraic Topology.) A boundary in a differential group... subgroup of Hol(D) For g ∈ Hol(D) and z ∈ D let j (g, z) be the determinant of the Jacobian transformation of g evaluated at z A meromorphic function f on D is an automorphic form of weight (an integer) for if f (γ z) = f (z)j (γ , z)− , γ ∈ , z ∈ D automorphism An isomorphism of a group, or algebra, onto itself See isomorphism automorphism group The set of all automorphisms of a group (vector space, algebra,. .. subgroup A subgroup H of a group G such that H is a maximal nilpotent subgroup of G and, for every subgroup K of H of finite index in H , the normalizer of K in G is also of finite index in K c 2001 by CRC Press LLC Cartier divisor A divisor which is linearly equivalent to the divisor 0 on a neighborhood of each point of an irreducible variety V Casimir element Let β1 , , βn be a basis of the semisimple . W αβ and W αβ such that the composition of the birational transformations between sub- sets of V α and V β and between subsets of V β and V γ are consistent with those between sub- sets of V α and. Engineers and Scientists Emma Previato The Comprehensive Dictionary of Mathematics Douglas N. Clark c  2001 by CRC Press LLC a Volume in the Comprehensive Dictionary of Mathematics DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY Edited. DICTIONARY OF ALGEBRA, ARITHMETIC, AND TRIGONOMETRY c  2001 by CRC Press LLC Comprehensive Dictionary of Mathematics Douglas N. Clark Editor-in-Chief Stan