User:Richard Pinch/sandbox

Exponentiation

The algebraic and analytic operations generalising the operation of repeated multiplication in number systems.

For positive integer $n$, the operation $x \mapsto x^n$ may be defined on any system of numbers by repeated multiplication $$ x^n = x \cdot x \cdot \cdots \cdot x\ \ \ (n\,\text{times}) $$ where $\cdot denotes multiplication. The number $n$ is the ''exponent'' in this operation. The repeated operations may be carried out in any order provided that multiplication is [[associativity|associative]], $x \cdot (y \cdot z) = (x \cdot y) \cdot z$. In this case we have '"`UNIQ-MathJax3-QINU`"' If the operation is also [[commutativity|commutative]] then we have '"`UNIQ-MathJax4-QINU`"' We may extend the definition to non-positive integer powers by defining $x^0 = 1$ and '"`UNIQ-MathJax5-QINU`"' whenever this makes sense. We may extend the definition to rational numbers by taking $x^{1/n}$ to be any number $y$ such that $y^n = x$: this may denote none, one or more than one number. ==='"`UNIQ--h-1--QINU`"'Positive real numbers=== Exponentiation of positive real numbers may be defined by taking $x^{1/n}$ to be the unique positive real solution of $y^n = x$: this always exists. We thus have $x^q$ well-defined for $x>0$ and any rational exponent $q$. Exponentiation preserves order: if $x > y$ then $x^q > y^q$ if $q > 0$ and $x^q < y^q$ if $q < 0$. We can now define exponentiation with real exponent $r$ by defining $x^r$ to be the limit of $x^{q_n}$ where $q_n$ is a sequence of rational numbers converging to $r$. There is always such a sequence, and the limit exists and does not depend on the chosen sequence. For positive real numbers there are mutually inverse [[exponential function, real|exponential]] and [[logarithmic function|logarithm]] functions which allow the alternative definition '"`UNIQ-MathJax6-QINU`"' Here the exponential function may be regarded as $\exp(x) = e^x$ where [[E-number|$e$]] is the base of natural logarithms. ==='"`UNIQ--h-2--QINU`"'General algebraic systems=== For a general (not necessarily associative) binary operation, it is necessary to define the order of operations. The left and right ''principal powers'' are defined inductively by '"`UNIQ-MathJax7-QINU`"' and '"`UNIQ-MathJax8-QINU`"' respectively. A binary operation is [[power associativity|power associative]] if the powers of a single element form an associative subsystem, so that exponentiation is well-defined. ='"`UNIQ--h-3--QINU`"'Logarithm= The operation inverse to [[exponentiation]]. Over the fields of [[Real number|real]] or [[complex number]]s, one speaks of the [[logarithm of a number]]. The [[logarithmic function]] is the complex analytic function inverse to the [[exponential function]]. In a finite Abelian group, the [[discrete logarithm]] is the inverse to exponentiation, with applications in [[cryptography]]. The [[Zech logarithm]] in a finite field is related to the discrete logarithm. ='"`UNIQ--h-4--QINU`"'''I''-semigroup= A topological semigroup defined on a totally ordered set. Let $I$ be a [[totally ordered set]] with minimum element $0$ and maximum element $1$, and equipped with the [[order topology]]; then $0$ acts as a zero (absorbing) element for the semigroup operation and $1$ acts as an identity (neutral) element. Although not required by the definition, it is the case that an ''I''-semigroup is commutative. Examples. The real interval $[0,1]$ under multiplication. The ''nil interval'' $[\frac12,1]$ with operation $x \circ y = \max(xy,\frac12)$. The ''min interval'' $[0,1]$ with operation $x \cdot y = \min(x,y)$. ===='"`UNIQ--h-5--QINU`"'References==== * Hofmann, K.H.; Lawson, J.D. "Linearly ordered semigroups: Historical origins and A. H. Clifford’s influence" ''in'' Hofmann, Karl H. (ed.) et al., ''Semigroup theory and its applications. Proceedings of the 1994 conference commemorating the work of Alfred H. Clifford'' London Math. Soc. Lecture Note Series '''231''' Cambridge University Press (1996) pp.15-39 [https://zbmath.org/?q=an%3A0901.06012 Zbl 0901.06012] ='"`UNIQ--h-6--QINU`"'Composition algebra= An algebra $A$ (not necessarily associative) over a field $K$ with a [[quadratic form]] $q$ taking values in $K$ which is multiplicative, $q(x\cdot y) = q(x) q(y)$. The composition algebras over the field $\mathbf{R}$ of [[real number]]s are the real numbers, the field of [[complex number]]s $\mathbf{C}$, the [[skew-field]] of [[quaternion]]s, the non-associative algebra of [[octonions]]. ===='"`UNIQ--h-7--QINU`"'References==== * Springer, Tonny A.; Veldkamp, Ferdinand D. ''Octonions, Jordan algebras and exceptional groups''. Springer Monographs in Mathematics. Springer (2000) ISBN 3-540-66337-1 [https://zbmath.org/?q=an%3A1087.17001 Zbl 1087.17001] ='"`UNIQ--h-8--QINU`"'Cayley–Dickson process= A construction of an algebra $A_1$ from an algebra $A$ with involution over a field $K$ which generalises the construction of the [[complex number]]s, [[quaternion]]s and [[octonion]]. Fix a parameter $d \in A$. As a set $A_1 = A \times A$ with addition defined by $(a_1,a_2) + (b_1,b_2) = (a_1+b_1, a_2+b_2)$ and multiplication by '"`UNIQ-MathJax9-QINU`"' The algebra $A_1$ has an involution $(x_1,x_2) \mapsto (x_1^*,-x_2)$. ='"`UNIQ--h-9--QINU`"'BCH code= A [[cyclic code]] over a finite field. Fix length $n$ and ground field $\mathbf{F}_q$ and a design distance parameter $\delta$. Let $\beta$ be a primitive $n$-th root of unity in a suitable extension of $\mathbf{F}_q$. The generator of the cyclic code is the least common multiple $g$ of the minimal polynomials (over $\mathbf{F}_q$) of the elements $\beta^1, \beta^2, \ldots, \beta^{\delta-1}$. The minimum distance of the BCH code generated by $g$ is at least $\delta$: this is the ''BCH bound''. As an example, let $q=2$ and $\beta$ be a primitive $7$-th root of unity in $\mathrm{F}_{8}$: we may take $\beta$ to satisfy the polynomial $x^3 + x + 1$. Choose $\delta = 3$. The minimal polynomial for $\beta^2$ is the same as that of $\beta$, so that the cyclic code is generated by the word $1101000$. This is in fact the Hamming [7,4] code. ===='"`UNIQ--h-10--QINU`"'References==== * Goldie, Charles M.; Pinch, Richard G.E. ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) iSBN 0-521-40456-8 [https://zbmath.org/?q=an%3A0746.94001 Zbl 0746.94001] ='"`UNIQ--h-11--QINU`"'Hamming distance= A function on words of fixed length over an alphabet describing the number of changes to the symbols of one word required to reduce it to another. Let $A$ be an alphabet of symbols and $C$ a subset of $A^n$, the set of words of length $n$ over $A$. Let $u=(u_1,\ldots,u_n)$ and $v = (v_1,\ldots,v_n)$ be words in $C$. The Hamming distance $d(u,v)$ is defined as the number of places in which $u$ and $v$ differ: that is, $\sharp\{ i : u_i \neq v_i,\,i=1,\dots,n \}$. The Hamming distance satisfies '"`UNIQ-MathJax10-QINU`"' '"`UNIQ-MathJax11-QINU`"' '"`UNIQ-MathJax12-QINU`"' Hamming distance is thus a [[metric]] on $C$. In the theory of [[error-correcting code]]s it is assumed that words from $C$ are transmitted down a noisy channel and that some symbols are changed during the transmission: see for example the [[symmetric channel]]. Maximum likelihood decoding of a received word $r$ consists of finding the word in $C$ nearest to $r$ in the Hamming metric. If the minimum Hamming distance between words of $C$ is $\delta$, then this process is capable of detecting up to $\delta-1$ errors in $r$, and correcting up to $\left\lfloor \frac{\delta-1}{2} \right\rfloor$ errors. ===='"`UNIQ--h-12--QINU`"'References==== * Goldie, Charles M.; Pinch, Richard G.E. ''Communication theory'', London Mathematical Society Student Texts. '''20''' Cambridge University Press (1991) iSBN 0-521-40456-8 [https://zbmath.org/?q=an%3A0746.94001 Zbl 0746.94001] ='"`UNIQ--h-13--QINU`"'Leibniz algebra= An algebra over a field $K$ generalising the properties of a [[Lie algebra]]. A Leibniz algebra $L$ is a $K$-algebra with multiplication denoted by $[\cdot,\cdot]$ satisfying '"`UNIQ-MathJax13-QINU`"' Every Lie algebra is a Leibniz algebra, and a Leibniz algebra is a Lie aglebra if in addition $[x,x] = 0$. The free Leibniz algebra on a generating set $X$ may be defined as the quotient of the free non-associative algebra over $K$ (cf. [[Free algebra over a ring]]) by the ideal generated by all elements of the form $[ x, [y,z]] - [[x,y],z] + [[x,z],y] $. The standard Leibniz algebra on $X$ is obtained from the vector space $V = KX$ and forming the tensor module '"`UNIQ-MathJax14-QINU`"' with the multiplication '"`UNIQ-MathJax15-QINU`"' when $v \in V$ and '"`UNIQ-MathJax16-QINU`"' The standard algebra is then a presentation of the free algebra on $X$. ====References==== * Mikhalev, Alexander A.; Shpilrain, Vladimir; Yu, Jie-Tai, ''Combinatorial methods. Free groups, polynomials, and free algebras'', CMS Books in Mathematics '''19''' Springer (2004) ISBN 0-387-40562-3 [https://zbmath.org/?q=an%3A1039.16024 Zbl 1039.16024] =Tridiagonal matrix= A matrix with non-zero entries only on the main diagonal and the diagonals immediately above and below, for example '"`UNIQ-MathJax17-QINU`"' The [[determinant]] of a tridiagonal matrix may be computed as a [[continuant]]. ==References== * Thomas Muir. ''A treatise on the theory of determinants''. (Dover Publications, 1960 [1933]) =Free differential calculus= Let $F$ be a [[free group]] on a set of generators $X = \{x_i : i \in I \}$ and $R[F]$ the [[group ring]] of $F$ over a commutative unital ring $R$. The ''Fox derivative'' $\partial_i$ are maps from $F$ to $R[F]$ defined by '"`UNIQ-MathJax18-QINU`"' '"`UNIQ-MathJax19-QINU`"' '"`UNIQ-MathJax20-QINU`"' It follows that '"`UNIQ-MathJax21-QINU`"' The maps extend to [[derivation]]s on $R[F]$. ===References=== * D. L. Johnson, ''Presentations of Groups'', London Mathematical Society Student Texts '''15''' Cambridge University Press (1997) ISBN 0-521-58542-2 =Martin's axiom= An axiom of set theory. Let $(P,{<})$ be a [[partially ordered set]] satisfying the [[countable chain condition]] and $D$ a family of $\mathfrak{k}$ dense subsets of $P$ for $\mathfrak{k}$ a cardinal less than $2^{\aleph_0}$. Then $\text{MA}_{\mathfrak{k}}$ asserts that there is a $D$-generic filter on $P$. Martin's axiom $\text{MA}$ is the conjunction of $\text{MA}_{\mathfrak{k}}$ for all $\mathfrak{k} < 2^{\aleph_0}$. The case $\text{MA}_{\aleph_0}$ holds in [[ZFC]]. MA is a consequence of the [[Continuum hypothesis]] ($\text{CH}$) but $\text{MA} \wedge \text{CH}$ is consistent with ZFC if ZFC is consistent. ===='"`UNIQ--h-14--QINU`"'References==== * Thomas Jech, ''Set Theory'', Perspectives in Mathematical Logic, Third Millennium Edition, revised and expanded. Springer (2007) ISBN 3-540-44761-X ='"`UNIQ--h-15--QINU`"'Schroeder–Bernstein theorem= ''Cantor–Bernstein theorem'' For sets $A$ and $B$, if there are [[injection]]s from $A$ to $B$ and from $B$ to $A$ (equivalently, each is equipotent to a subset of the other), then there is a bijection between $A$ and $B$ (they are equipotent). In cardinal arithmetic, if we let $\mathfrak{a} \le \mathfrak{b}$ denote the property that some set of cardinality $\mathfrak{a}$ has an injection to a set of cardinality $\mathfrak{b}$, then $\mathfrak{a} \le \mathfrak{b}$ and $\mathfrak{b} \le \mathfrak{a}$ implies $\mathfrak{a} = \mathfrak{b}$. The theorem was conjectured by Georg Cantor by 1895 and proved by Felix Bernstein in 1897. Dedekind obtained a further proof in 1897. Schroeder proof of 1898 was found to be flawed by 1902. ===='"`UNIQ--h-16--QINU`"'References==== * P. R. Halmos, "Naive Set Theory", Springer (1960) ISBN 0-387-90092-6 * Michael Potter, "Set Theory and its Philosophy : A Critical Introduction", Oxford University Press (2004) ISBN 0-19-155643-2 ='"`UNIQ--h-17--QINU`"'Hurwitz zeta function= ''generalised zeta function'' An [[Dirichlet series]] related to the [[Riemann zeta function]] which may be used to exhibit properties of various [[Dirichlet L-function]]s. The Hurwitz zeta function $\zeta(\alpha,s)$ is defined for real $\alpha$, $0 < \alpha \le 1$ as '"`UNIQ-MathJax22-QINU`"' The series is convergent, and defines an analytic function, for $\Re s > 1$. The function possesses an [[analytic continuation]] to the whole $s$-plane except for a simple pole of residue 1 at $s=1$. =='"`UNIQ--h-18--QINU`"'References== * Tom M. Apostol, "Introduction to Analytic Number Theory", Undergraduate Texts in Mathematics, Springer (1976) ISBN 0-387-90163-9 [https://zbmath.org/?q=an%3A0335.10001 Zbl 0335.10001] ='"`UNIQ--h-19--QINU`"'Isomorphism theorems= Three theorems relating to [[homomorphism]]s of general [[algebraic system]]s. =='"`UNIQ--h-20--QINU`"'First Isomorphism Theorem== Let $f : A \rightarrow B$ be a [[homomorphism]] of $\Omega$-algebras and $Q$ the [[Kernel of a function|kernel]] of $f$, as an [[equivalence relation]] on $A$. Then $Q$ is a [[Congruence (in algebra)|congruence]] on $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/Q$ is the quotient map, $f' : A/Q \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism. =='"`UNIQ--h-21--QINU`"'Second Isomorphism Theorem== Let $Q$ be a congruence on the $\Omega$-algebra $A$ and let $A_1$ be a subalgebra of $A$. The saturation $A_1^Q$ is a subalgebra of $A$, the restriction $Q_1 = Q \ \cap A_1 \times A_1$ is a congruence on $A_1$ and there is an isomorphism '"`UNIQ-MathJax23-QINU`"' =='"`UNIQ--h-22--QINU`"'Third Isomorphism Theorem== Let $A$ be an $\Omega$-algebra and $Q \subset R$ congruences on $A$. There is a unique homomorphism $\theta$ from $A/Q \rightarrow A/R$ compatible with the quotient maps from $A$ to $A/R$ and $A/Q$. If $R/Q$ denotes the [[Kernel of a function|kernel]] of $\theta$ on $A/Q$ then there is an isomorphism '"`UNIQ-MathJax24-QINU`"' =='"`UNIQ--h-23--QINU`"'Application to groups== In the case of [[group]]s, a congruence $Q$ on $G$ is determined by the congruence class $N = [1_G]_Q$ of the identity $1_G$, which is a [[normal subgroup]], and the other $Q$-classes are the cosets of $N$. It is conventional to write $G/N$ for $G/Q$. The saturation of a subgroup $H$ is the complex $H^Q = HN$. ==='"`UNIQ--h-24--QINU`"'First Isomorphism Theorem for groups=== Let $f : A \rightarrow B$ be a [[homomorphism]] of groups and $N = f^{-1}(1_B)$ the [[Kernel of a morphism in a category|kernel]] of $f$. Then $N$ is a normal subgroup of $A$, and $f$ can be factorised as $f = \epsilon f' \mu$ where $\epsilon : A \rightarrow A/N$ is the quotient map, $f' : A/N \rightarrow f(A)$ and $\mu : f(A) \rightarrow B$ is the inclusion map. The theorem asserts that $f'$ is well-defined and an isomorphism. ==='"`UNIQ--h-25--QINU`"'Second Isomorphism Theorem for groups=== Let $N$ be a normal subgroup $A$ and let $A_1$ be a subgroup of $A$. The complex $NA_1$ is a subgroup of $A$, the intersection $N_1 = N \cap A_1$ is a normal subgroup o $A_1$ and there is an isomorphism '"`UNIQ-MathJax25-QINU`"' ==='"`UNIQ--h-26--QINU`"'Third Isomorphism Theorem for groups=== Let $A$ be a group and $N \subset M$ normal subgroups of $A$. There is a unique homomorphism $\theta$ from $A/N \rightarrow A/M$ compatible with the quotient maps from $A$ to $A/N$ and $A/M$. The set $M/N$ is the kernel of $\theta$ and hence a normal subgroup of $A/N$ and there is an isomorphism '"`UNIQ-MathJax26-QINU`"' =='"`UNIQ--h-27--QINU`"'References== * Paul M. Cohn, ''Universal algebra'', Kluwer (1981) ISBN 90-277-1213-1 ='"`UNIQ--h-28--QINU`"'Dilworth number= A numerical invariant of a [[graph]] (cf. [[Graph, numerical characteristics of a]]). The Dilworth number of the graph $G$ is the maximum number of vertices in a set $D$ for which the neighbourhoods form a [[Sperner family]]: for any pair of such vertices $x,y \in D$, the neighbourhoods $N(x)$, $N(y)$ each has at least one element not contained in the other. It is the [[Width of a partially ordered set|width]] of the set of neighbourhoods partially ordered by inclusion. The [[diameter]] of a graph exceeds its Dilworth number by at most 1. There is a polynomial time algorithm for computing the Dilworth number of a finite graph. =='"`UNIQ--h-29--QINU`"'References== * Andreas Brandstädt, Van Bang Le; Jeremy P. Spinrad, "Graph classes: a survey". SIAM Monographs on Discrete Mathematics and Applications '''3'''. Society for Industrial and Applied Mathematics (1999) ISBN 978-0-898714-32-6 [https://zbmath.org/?q=an%3A0919.05001 Zbl 0919.05001] ='"`UNIQ--h-30--QINU`"'Square-free= ''quadratfrei'' Containing only a trivial square factor. A natural number $n$ is square-free if the only natural number $d$ such that $d^2$ divides $n$ is $d=1$. The prime power factorisation of such a number $n$ has all exponents equal to 1. Similarly a polynomial $f$ is square-free if the only factors $g$ such that $g^2$ divides $f$ are constants. For polynomials, this is equivalent to having no repeated roots in any field. An element $x$ of a [[monoid]] $M$ is square-free if the only $y \in M$ such that $y^2$ divides $x$ are units. A word $x$ over an alphabet $A$, that is, an element of the free monoid $A^*$, is square-free if $x=uwwv$ implies that $w$ is the empty string. =='"`UNIQ--h-31--QINU`"'Square-free number== An integer $n$ is square-free if the only natural number $d$ such that $d^2$ divides $n$ is $d=1$. The prime factorisation of such a number $n$ has all exponents equal to 1. Any integer is uniquely expressible in the form $n = k^2 m$ where $m$ is the ''square-free kernel'' of $n$. If $Q(x)$ counts the square-free natural numbers $\le x$, then '"`UNIQ-MathJax27-QINU`"' ==='"`UNIQ--h-32--QINU`"'References=== * E. Landau, "Über den Zusammenhang einiger neuerer Sätze der analytischen Zahlentheorie", Wien. Ber. '''115''' (1906) 589-632. [https://zbmath.org/?q=an%3A37.0236.01 Zbl 37.0236.01] * József Sándor; Dragoslav S. Mitrinović; Borislav Crstici, edd. "Handbook of number theory I". Springer-Verlag (2006). Sect.VI.18. ISBN 1-4020-4215-9. [https://zbmath.org/?q=an%3A1151.11300 Zbl 1151.11300] =='"`UNIQ--h-33--QINU`"'Square-free polynomial== A polynomial $f$ over a field is square-free if the only factors $g$ such that $g^2$ divides $f$ are constants. For polynomials, this is equivalent to having no repeated roots. Over fields of [[Characteristic of a field|characteristic]] zero, a polynomial is square-free if and only if it is coprime to its formal derivative. Over fields of characteristic $p$, this holds for [[separable polynomial]]s, those $f$ such that $f' \not\equiv 0$, that is, those polynomials in $X$ that are not polynomials in $X^p$. Over a finite field $\mathbb{F}_q$, the number of square-free monic polynomials of degree $d$ is $(1-q^{-1})q^d$. ==='"`UNIQ--h-34--QINU`"'References=== * Gary L. Mullen, Daniel Panario (edd), "Handbook of Finite Fields", CRC Press (2013) ISBN 1439873828 =='"`UNIQ--h-35--QINU`"'Square-free word== A word $x$ over an alphabet $A$, that is, an element of the free monoid $A^*$, is square-free if $x=uwwv$ implies that $w$ is the empty string. A square-free word is thus one that [[Avoidable pattern|avoids the pattern]] $XX$.'"`UNIQ--ref-00000000-QINU`"''"`UNIQ--ref-00000001-QINU`"' ==='"`UNIQ--h-36--QINU`"'Examples=== Over a two-letter alphabet $\{a,b\}$ the only square-free words are the empty word and $a$, $b$, $ab$, $ba$m $aba$ and $bab$. However, there exist infinite square-free words in any [[alphabet]] with three or more symbols,'"`UNIQ--ref-00000002-QINU`"' as proved by Axel Thue.'"`UNIQ--ref-00000003-QINU`"''"`UNIQ--ref-00000004-QINU`"' One example of an infinite square-free word over an alphabet of size 3 is the word over the alphabet $\{0,\pm1\}$ obtained by taking the [[first difference]] of the [[Thue–Morse sequence]].'"`UNIQ--ref-00000005-QINU`"''"`UNIQ--ref-00000006-QINU`"' An example found by John Leech'"`UNIQ--ref-00000007-QINU`"' is defined recursively over the alphabet $\{a,b,c\}$. Let \(w_1\) be any word starting with the letter $a$. Define the words \( \{w_i \mid i \in \mathbb{N} \}\) recursively as follows: the word \(w_{i+1}\) is obtained from \(w_i\) by replacing each $a$ in \(w_i\) with $abcbacbcabcba$, each $b$ with $bcacbacabcacb$, and each $c$ with $cabacbabcabac$. The sequence $(w_i)$ converges to the infinite square-free word '"`UNIQ-MathJax28-QINU`"' ==='"`UNIQ--h-37--QINU`"'Related concepts=== A '''cube-free''' word is one with no occurrence of $www$ for a factor $w$. The Thue–Morse sequence is an example of a cube-free word over a binary alphabet.'"`UNIQ--ref-00000008-QINU`"' This sequence is not square-free but is "almost" so: the [[Critical exponent of a word|critical exponent]] is 2.'"`UNIQ--ref-00000009-QINU`"' The Thue–Morse sequence has no '''overlap''' or ''overlapping square'', instances of $0X0X0$ or $1X1X1$:'"`UNIQ--ref-0000000A-QINU`"' it is essentially the only infinite binary word with this property.'"`UNIQ--ref-0000000B-QINU`"' An '''abelian $p$-th power''' is a subsequence of the form \(w_1 \cdots w_p\) where each \(w_i\) is a permutation of \(w_1\). There is no abelian-square-free infinite word over an alphabet of size three: indeed, every word of length eight over such an alphabet contains an abelian square. There is an infinite abelian-square-free word over an alphabet of size five.'"`UNIQ--ref-0000000C-QINU`"' ==='"`UNIQ--h-38--QINU`"'References=== '"`UNIQ--references-0000000D-QINU`"' * Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V.; <i>Combinatorics on words. Christoffel words and repetitions in words</i>, <i>ser.</i> CRM Monograph Series '''27''' (2009), American Mathematical Society, [https://zbmath.org/?q=an%3A1161.68043 Zbl 1161.68043]URL: http://www.ams.org/bookpages/crmm-27] ISBN: 978-0-8218-4480-9 * Lothaire, M.; <i>Combinatorics on words</i>, (1997), Cambridge University Press ISBN: 0-521-59924-5. * Lothaire, M.; <i>Algebraic combinatorics on words</i>, <i>ser.</i> Encyclopedia of Mathematics and Its Applications '''90''' (2011), Cambridge University Press, [https://zbmath.org/?q=an%3A1221.68183 Zbl 1221.68183] ISBN: 978-0-521-18071-9 * Pytheas Fogg, N.; <i>Substitutions in dynamics, arithmetics and combinatorics</i>, <i>ser.</i> Lecture Notes in Mathematics '''1794''' (2002), Springer-Verlag, [https://zbmath.org/?q=an%3A1014.11015 Zbl 1014.11015] ISBN: 3-540-44141-7 ='"`UNIQ--h-39--QINU`"'Root of unity= An element $\zeta$ of a ring $R$ with unity $1$ such that $\zeta^m = 1$ for some $m \ge 1$. The least such $m$ is the order of $\zeta$. A primitive root of unity of order $m$ is an element $\zeta$ such that $\zeta^m = 1$ and $\zeta^r \neq 1$ for any positive integer $r < m$. The element $\zeta$ generates the [[cyclic group]] $\mu_m$ of roots of unity of order $m$. For any $k$ that is relatively prime to $m$, the element $\zeta^k$ is also a primitive root. The number of all primitive roots of order $m$ is equal to the value of the [[Euler function]] $\phi(m)$ if $\mathrm{hcf}(m,\mathrm{char}(K)) = 1$. The primitive roots of unity are the roots of the [[cyclotomic polynomial]] of order $m$. If the field $K$ contains a primitive root of unity of order $m$, then $m$ is relatively prime to the [[Characteristic of a field|characteristic]] of $K$. An [[algebraically closed field]] contains a primitive root of any order that is relatively prime with its characteristic. In the field of [[complex number]]s, there are roots of unity of every order: those of order $m$ take the form '"`UNIQ-MathJax29-QINU`"' where $0 < k < m$ and $k$ is relatively prime to $m$. ===='"`UNIQ--h-40--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> S. Lang, "Algebra" , Addison-Wesley (1984)</td></tr> </table> ='"`UNIQ--h-41--QINU`"'Law of quadratic reciprocity= =='"`UNIQ--h-42--QINU`"'Gauss reciprocity law== A relation connecting the values of the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. [[Quadratic reciprocity law|Quadratic reciprocity law]]). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation '"`UNIQ-MathJax30-QINU`"' there are two more additions to this law, viz.: '"`UNIQ-MathJax31-QINU`"' The reciprocity law for quadratic residues was first stated in 1772 by L. Euler. A. Legendre in 1785 formulated the law in modern form and proved a part of it. C.F. Gauss in 1801 was the first to give a complete proof of the law [[#References|[1]]]; he also gave no less than eight different proofs of the reciprocity law, based on various principles, during his lifetime. Attempts to establish the reciprocity law for [[Cubic residue|cubic]] and [[biquadratic residue]]s led Gauss to introduce the ring of [[Gaussian integer]]s. ===='"`UNIQ--h-43--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)</td></tr> <tr><td valign="top">[2]</td> <td valign="top"> I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)</td></tr> <tr><td valign="top">[3]</td> <td valign="top"> H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)</td></tr> </table> ===='"`UNIQ--h-44--QINU`"'Comments==== Attempts to generalize the quadratic reciprocity law (as Gauss' reciprocity law is usually called) have been an important driving force for the development of [[algebraic number theory]] and [[class field theory]]. A far-reaching generalization of the quadratic reciprocity law is known as Artin's reciprocity law. =='"`UNIQ--h-45--QINU`"'Quadratic reciprocity law== The relation '"`UNIQ-MathJax32-QINU`"' connecting the Legendre symbols (cf. [[Legendre symbol|Legendre symbol]]) '"`UNIQ-MathJax33-QINU`"' for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, namely: '"`UNIQ-MathJax34-QINU`"' and '"`UNIQ-MathJax35-QINU`"' C.F. Gauss gave the first complete proof of the quadratic reciprocity law, which for this reason is also called the [[Gauss reciprocity law|Gauss reciprocity law]]. It immediately follows from this law that for a given square-free number $d$, the primes $p$ for which $d$ is a quadratic residue modulo $p$ ly in certain arithmetic progressions with common difference $2|d|$ or $4|d|$. The number of these progressions is $\phi(2|d|)/2$ or $\phi(4|d|)/2$, where $\phi(n)$ is the [[Euler function|Euler function]]. The quadratic reciprocity law makes it possible to establish factorization laws in quadratic extensions $\mathbf Q(\sqrt d)$ of the field of rational numbers, since the factorization into prime factors in $\mathbf Q(\sqrt d)$ of a prime number that does not divide $d$ depends on whether or not $x^2-d$ is reducible modulo $p$. ===='"`UNIQ--h-46--QINU`"'References==== <table><tr><td valign="top">[1]</td> <td valign="top"> I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)</td></tr><tr><td valign="top">[2]</td> <td valign="top"> Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)</td></tr></table> ===='"`UNIQ--h-47--QINU`"'Comments==== See also [[Quadratic residue|Quadratic residue]]; [[Dirichlet character|Dirichlet character]]. ===='"`UNIQ--h-48--QINU`"'References==== <table><tr><td valign="top">[a1]</td> <td valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII</td></tr></table> ='"`UNIQ--h-49--QINU`"'Euler function= =='"`UNIQ--h-50--QINU`"'Euler function== The arithmetic function $\phi$ whose value at $n$ is equal to the number of positive integers not exceeding $n$ and relatively prime to $n$ (the "totatives" of $n$). The Euler function is a [[multiplicative arithmetic function]], that is $\phi(1)=1$ and $\phi(mn)=\phi(m)\phi(n)$ for $(m,n)=1$. The function $\phi(n)$ satisfies the relations '"`UNIQ-MathJax36-QINU`"' '"`UNIQ-MathJax37-QINU`"' '"`UNIQ-MathJax38-QINU`"' It was introduced by L. Euler (1763). The function $\phi(n)$ can be evaluated by $\phi(n)=n\prod_{p|n}(1-p^{-1})$, where the product is taken over all primes dividing $n$, cf. [[#References|[a1]]]. For a derivation of the asymptotic formula in the article above, as well as of the formula '"`UNIQ-MathJax39-QINU`"' where $\gamma$ is the [[Euler constant]], see also [[#References|[a1]]], Chapts. 18.4 and 18.5. The Carmichael conjecture on the Euler totient function states that if $\phi(x) = m$ for some $m$, then $\phi(y) = m$ for some $y \neq x$; i.e. no value of the Euler function is assumed once. This has been verified for $x < 10^{1000000}$, [[#References|[c1]]]. D. H. Lehmer asked whether whether there is any composite number $n$ such that $\phi(n)$ divides $n-1$. This is true of every [[prime number]], and Lehmer conjectured in 1932 that there are no composite numbers with this property: he showed that if any such $n$ exists, it must be odd, square-free, and divisible by at least seven primes [[#References|[c2]]]. For some results on this still (1996) largely open problem, see [[#References|[c3]]] and the references therein. ===='"`UNIQ--h-51--QINU`"'References==== <table> <tr><td valign="top">[1]</td> <td valign="top"> K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) [https://mathscinet.ams.org/mathscinet/article?mr=0249348 MR0249348] [https://zbmath.org/?q=an%3A0169.37502 Zbl 0169.37502]</td></tr> <tr><td valign="top">[a1]</td> <td valign="top"> G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 [https://mathscinet.ams.org/mathscinet/article?mr=0568909 MR0568909] [https://zbmath.org/?q=an%3A0423.10001 Zbl 0423.10001]</td></tr> <tr><td valign="top">[c1]</td> <td valign="top"> A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" ''Math. Comp.'' , '''63''' (1994) pp. 415–419</td></tr> <tr><td valign="top">[c2]</td> <td valign="top"> D.H. Lehmer, "On Euler's totient function", ''Bulletin of the American Mathematical Society'' '''38''' (1932) 745–751 [https://doi.org/10.1090/s0002-9904-1932-05521-5 DOI 10.1090/s0002-9904-1932-05521-5] [https://zbmath.org/?q=an%3A0005.34302 Zbl 0005.34302]</td></tr> <tr><td valign="top">[c3]</td> <td valign="top"> V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" ''Nieuw Archief voor Wiskunde (4)'' , '''5''' (1987) pp. 77–83</td></tr> <tr><td valign="top">[c4]</td> <td valign="top"> M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" ''Rocky Mount. J. Math.'' , '''15''' (1985) pp. 609–620</td></tr> <tr><td valign="top">[c5]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" ''Nieuw Archief Wiskunde (4)'' , '''8''' (1990) pp. 169–188</td></tr> <tr><td valign="top">[c6]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient I" ''Nieuw Archief Wiskunde (4)'' , '''4''' (1986) pp. 175–190</td></tr> <tr><td valign="top">[c7]</td> <td valign="top"> L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155</td></tr> </table> =='"`UNIQ--h-52--QINU`"'Totient function== ''Euler totient function, Euler totient'' Another frequently used named for the [[Euler function]] $\phi(n)$, which counts a [[reduced system of residues]] modulo $n$: the natural numbers $k \in \{1,\ldots,n\}$ that are relatively prime to $n$. A natural generalization of the Euler totient function is the [[Jordan totient function]] $J_k(n)$, which counts the number of $k$-tuples $(a_1,\ldots,a_k)$, $a_i \in \{1,\ldots,n\}$, such that $\mathrm{hcf}\{n,a_1,\ldots,a_k\} = 1$. Clearly, $J_1 = \phi$. The $J_k$ are [[multiplicative arithmetic function]]s. One has '"`UNIQ-MathJax40-QINU`"' where $p$ runs over the prime numbers dividing $n$, and '"`UNIQ-MathJax41-QINU`"' where $\mu$ is the [[Möbius function]] and $d$ runs over all divisors of $n$. For $k=1$ these formulae reduce to the well-known formulae for the Euler function. ===='"`UNIQ--h-53--QINU`"'References==== <table> <tr><td valign="top">[a1]</td> <td valign="top"> A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" ''Math. Comp.'' , '''63''' (1994) pp. 415–419</td></tr> <tr><td valign="top">[a2]</td> <td valign="top"> D.H. Lehmer, "On Euler's totient function" ''Bull. Amer. Math. Soc.'' , '''38''' (1932) pp. 745–751</td></tr> <tr><td valign="top">[a3]</td> <td valign="top"> V. Siva Rama Prasad, M. Rangamma, "On composite $n$ for which $\phi(n)|n-1$" ''Nieuw Archief voor Wiskunde (4)'' , '''5''' (1987) pp. 77–83</td></tr> <tr><td valign="top">[a4]</td> <td valign="top"> M.V. Subbarao, V. Siva Rama Prasad, "Some analogues of a Lehmer problem on the totient function" ''Rocky Mount. J. Math.'' , '''15''' (1985) pp. 609–620</td></tr> <tr><td valign="top">[a5]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" ''Nieuw Archief Wiskunde (4)'' , '''8''' (1990) pp. 169–188</td></tr> <tr><td valign="top">[a6]</td> <td valign="top"> R. Sivamarakrishnan, "The many facets of Euler's totient I" ''Nieuw Archief Wiskunde (4)'' , '''4''' (1986) pp. 175–190</td></tr> <tr><td valign="top">[a7]</td> <td valign="top"> L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155</td></tr> </table> =='"`UNIQ--h-54--QINU`"'Jordan totient function== An arithmetic function $J_k(n)$ of a [[natural number]] $n$, named after Camille Jordan, counting the $k$-tuples of positive integers all less than or equal to $n$ that form a [[Coprime numbers|coprime]] $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's [[totient function]], which is $J_1$. Jordan's totient function is [[Multiplicative arithmetic function|multiplicative]] and may be evaluated as '"`UNIQ-MathJax42-QINU`"' By [[Möbius inversion]] we have $\sum_{d | n } J_k(d) = n^k $. The [[Average order of an arithmetic function|average order]] of $J_k(n)$ is $c n^k$ for some $c$. The analogue of Lehmer's problem for the Jordan totient function (and $k>1$) is easy, [[#References|[c4]]]: For $k>1$, $J_k(n) | n^k-1$ if and only if $n$ is a prime number. Moreover, if $n$ is a prime number, then $J_k(n) = n^k-1$. For much more information on the Euler totient function, the Jordan totient function and various other generalizations, see [[#References|[c5]]], [[#References|[c6]]]. ===='"`UNIQ--h-55--QINU`"'References==== * Dickson, L.E. ''History of the Theory of Numbers I'', Chelsea (1971) p. 147, ISBN 0-8284-0086-5 * Ram Murty, M. ''Problems in Analytic Number Theory'', Graduate Texts in Mathematics '''206''' Springer-Verlag (2001) p. 11 ISBN 0387951431 [https://zbmath.org/?q=an%3A0971.11001 Zbl 0971.11001] * Sándor, Jozsef; Crstici, Borislav, edd. Handbook of number theory II. Dordrecht: Kluwer Academic (2004). pp.32–36. ISBN 1-4020-2546-7. [https://zbmath.org/?q=an%3A1079.11001 Zbl 1079.11001] ='"`UNIQ--h-56--QINU`"'Multiplicative sequence= Also ''m''-sequence, a sequence of [[polynomial]]s associated with a formal group structure. They have application in the [[cobordism|cobordism ring]] in [[algebraic topology]]. =='"`UNIQ--h-57--QINU`"'Definition== Let $K_n$ be polynomials over a ring $A$ in indeterminates $p_1,\ldots$ weighted so that $p_i$ has weight $i$ (with $p_0=1$) and all the terms in $K_n$ have weight $n$ (so that $K_n$ is a polynomial in $p_1,\ldots,p_n$). The sequence $K_n$ is ''multiplicative'' if an identity '"`UNIQ-MathJax43-QINU`"' implies '"`UNIQ-MathJax44-QINU`"' The power series '"`UNIQ-MathJax45-QINU`"' is the ''characteristic power series'' of the $K_n$. A multiplicative sequence is determined by its characteristic power series $Q(z)$, and every power series with constant term 1 gives rise to a multiplicative sequence. To recover a multiplicative sequence from a characteristic power series $Q(z)$ we consider the coefficient of ''z''^''j'' in the product '"`UNIQ-MathJax46-QINU`"' for any $m>j$. This is symmetric in the $\beta_i$ and homogeneous of weight ''j'': so can be expressed as a polynomial $K_j(p_1,\ldots,p_j)$ in the [[elementary symmetric function]]s $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence. =='"`UNIQ--h-58--QINU`"'Examples== As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$. Consider the power series '"`UNIQ-MathJax47-QINU`"' where $B_k$ is the $k$-th [[Bernoulli number]]. The multiplicative sequence with $Q$ as characteristic power series is denoted $L_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series '"`UNIQ-MathJax48-QINU`"' is denoted $A_j(p_1,\ldots,p_j)$. The multiplicative sequence with characteristic power series '"`UNIQ-MathJax49-QINU`"' is denoted $T_j(p_1,\ldots,p_j)$: the ''[[Todd polynomial]]s''. =='"`UNIQ--h-59--QINU`"'Genus== The '''genus''' of a multiplicative sequence is a [[ring homomorphism]], from the [[cobordism|cobordism ring]] of smooth oriented [[compact manifold]]s to another [[ring]], usually the ring of [[rational number]]s. For example, the [[Todd genus]] is associated to the Todd polynomials $T_j$ with characteristic power series '"`UNIQ-MathJax50-QINU`"' and the [[L-genus]] is associated to the polynomials $L_j$ with charac\teristic polynomial '"`UNIQ-MathJax51-QINU`"' =='"`UNIQ--h-60--QINU`"'References== * Hirzebruch, Friedrich. ''Topological methods in algebraic geometry'', Classics in Mathematics. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. Reprint of the 2nd, corr. print. of the 3rd ed. [1978] (Berlin: Springer-Verlag, 1995). ISBN 3-540-58663-6. [https://zbmath.org/?q=an%3A0843.14009 Zbl 0843.14009]. ='"`UNIQ--h-61--QINU`"'Nagao's theorem= A result, named after Hirosi Nagao, about the structure of the [[group]] of 2-by-2 [[Invertible matrix|invertible matrices]] over the [[ring of polynomials]] over a [[field]]. It has been extended by [[Jean-Pierre Serre|Serre]] to give a description of the structure of the corresponding matrix group over the [[coordinate ring]] of a [[projective curve]]. =='"`UNIQ--h-62--QINU`"'Nagao's theorem== For a [[Ring (mathematics)|general ring]] $R$ we let $GL_2(R)$ denote the group of invertible 2-by-2 matrices with entries in $R$, and let $R^*$ denote the [[group of units]] of $R$, and let '"`UNIQ-MathJax52-QINU`"' Then $B(R)$ is a subgroup of $GL_2(R)$. Nagao's theorem states that in the case that $R$ is the ring $K[t]$ of polynomials in one variable over a field $K$, the group $GL_2(R)$ is the [[amalgamated product]] of $GL_2(K)$ and $B(K[t])$ over their intersection $B(K)$. =='"`UNIQ--h-63--QINU`"'Serre's extension== In this setting, $C$ is a [[Singular point of an algebraic variety|smooth]] projective curve over a field $K$. For a [[closed point]] $P$ of $C$ let $R$ be the corresponding coordinate ring of $C$ with $P$ removed. There exists a [[graph of groups]] $(G,T)$ where $T$ is a [[tree]] with at most one non-terminal vertex, such that $GL_2(R)$ is isomorphic to the [[fundamental group]] $\pi_1(G,T)$. =='"`UNIQ--h-64--QINU`"'References== * Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". ''Transactions of the American Mathematical Society'' '''353''' (2001) 749–767. [https://doi.org/10.1090/S0002-9947-00-02707-0 DOI 10.1090/S0002-9947-00-02707-0] [https://zbmath.org/?q=an%3A0964.20027 Zbl 0964.20027]. * Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A '''10''' (1959) 117–121. [https://mathscinet.ams.org/mathscinet/article?mr=0114866 MR0114866]. [https://zbmath.org/?q=an%3A0092.02504 Zbl 0092.02504]. * Serre, Jean-Pierre. ''Trees''. (Springer, 2003) ISBN 3-540-44237-5. ='"`UNIQ--h-65--QINU`"'Brauer–Wall group= A [[group]] classifying graded [[central simple algebra]]s over a field. It was first defined by Wall (1964) as a generalisation of the [[Brauer group]]. The Brauer group $\mathrm{B}(F)$ of a field $F$ is defined on the isomorphism classes of central simple algebras over ''F''. The analogous construction for $\mathbf{Z}/2$-[[graded algebra]]s defines the Brauer–Wall group $\mathrm{BW}(F)$.[[#Lam (2005) pp.98–99|[Lam (2005) pp.98–99]]] =='"`UNIQ--h-66--QINU`"'Properties== * The Brauer group $\mathrm{B}(F)$ injects into $\mathrm{BW}(F)$ by mapping a CSA $A$ to the graded algebra which is $A$ in grade zero. There is an exact sequence '"`UNIQ-MathJax53-QINU`"' where $Q(F)$ is the group of graded quadratic extensions of $F$, defined as $\mathbf{Z}/2 \times F^*/(F^*)^2$ with multiplication $(e,x)(f,y) = (e+f,(-1)^{ef}xy)$. The map from W to BW is the '''[[Clifford invariant]]''' defined by mapping an algebra to the pair consisting of its grade and [[Determinant of a quadratic form|determinant]]. There is a map from the additive group of the [[Witt–Grothendieck ring]] to the Brauer–Wall group obtained by sending a quadratic space to its [[Clifford algebra]]. The map factors through the [[Witt group]][[#Lam (2005) p.113|[Lam (2005) p.113]]] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[[#Lam (2005) p.115|[Lam (2005) p.115]]] =='"`UNIQ--h-67--QINU`"'Examples== * $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of [[Bott periodicity]]. =='"`UNIQ--h-68--QINU`"'References== * Lam, Tsit-Yuen, ''Introduction to Quadratic Forms over Fields'', Graduate Studies in Mathematics '''67''', (American Mathematical Society, 2005) ISBN 0-8218-1095-2 [https://mathscinet.ams.org/mathscinet/article?mr=2104929 MR2104929], [https://zbmath.org/?q=an%3A1068.11023 Zbl 1068.11023] * Wall, C. T. C., "Graded Brauer groups", ''Journal für die reine und angewandte Mathematik'' '''213''' (1964) 187–199, ISSN 0075-4102, [https://zbmath.org/?q=an%3A0125.01904 Zbl 0125.01904], [https://mathscinet.ams.org/mathscinet/article?mr=0167498 MR0167498] ='"`UNIQ--h-69--QINU`"'Factor system= A function on a [[group]] giving the data required to construct an [[algebra]]. A factor system constitutes a realisation of the cocycles in the second [[cohomology group]] in [[group cohomology]]. Let $G$ be a group and $L$ a field on which $G$ acts as automorphisms. A ''cocycle'' or ''factor system'' is a map $c : G \times G \rightarrow L^*$ satisfying '"`UNIQ-MathJax54-QINU`"' Cocycles $c,c'$ are ''equivalent'' if there exists some system of elements $a : G \rightarrow L^*$ with '"`UNIQ-MathJax55-QINU`"' Cocycles of the form '"`UNIQ-MathJax56-QINU`"' are called ''split''. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$. =='"`UNIQ--h-70--QINU`"'Crossed product algebras== Let us take the case that $G$ is the [[Galois group]] of a [[field extension]] $L/K$. A factor system $c$ in $H^2(G,L^*)$ gives rise to a ''crossed product algebra'' $A$, which is a $K$-algebra containing $L$ as a subfield, generated by the elements $\lambda \in L$ and $u_g$ with multiplication '"`UNIQ-MathJax57-QINU`"' '"`UNIQ-MathJax58-QINU`"' Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write '"`UNIQ-MathJax59-QINU`"' Every [[central simple algebra]] over$K$ that splits over $L$ arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements in$H^2$. We thus obtain an identification of the [[Brauer group]], where the elements are classes of CSAs over $K$, with $H^2$.[[#Saltman (1999) p.44|[Saltman (1999) p.44]]] =='"`UNIQ--h-71--QINU`"'Cyclic algebra== Let us further restrict to the case that $L/K$ is [[Cyclic extension|cyclic]] with Galois group $G$ of order $n$ generated by $t$. Let $A$ be a crossed product $(L,G,c)$ with factor set $c$. Let $u=u_t$ be the generator in $A$ corresponding to $t$. We can define the other generators '"`UNIQ-MathJax60-QINU`"' and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by '"`UNIQ-MathJax61-QINU`"' It thus makes sense to denote $A$ simply by $(L,t,a)$. However $a$ is not uniquely specified by $A$ since we can multiply $u$ by any element $\lambda$ of $L^*$ and then $a$ is multiplied by the product of the conjugates of λ. Hence $A$ corresponds to an element of the norm residue group $K^*/N_{L/K}L^*$. We obtain the isomorphisms $$ \mathop{Br}(L/K) \equiv K^*/\mathrm{N}_{L/K} L^* \equiv \mathrm{H}^2(G,L^*) \ . $$

References

• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Universitext. Translated from the German by Silvio Levy. With the collaboration of the translator. Springer-Verlag. ISBN 978-0-387-72487-4. Zbl 1130.12001.
• Saltman, David J. (1999). Lectures on division algebras. Regional Conference Series in Mathematics 94. Providence, RI: American Mathematical Society. ISBN 0-8218-0979-2. Zbl 0934.16013.