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Quaternion algebra

An associative algebra over a field that generalises the construction of the quaternions over the field of real numbers.

The quaternion algebra $(a,b)_F$ over a field $F$ is the four-dimensional vector space with basis $\mathbf{1}, \mathbf{i}, \mathbf{j}, \mathbf{k}$ and multiplication defined by $$ \mathbf{i}^2 = a\mathbf{1},\ \ \mathbf{j}^2 = b\mathbf{1},\ \ \mathbf{i}\mathbf{j} = -mathbf{j}\mathbf{i} = \mathbf{k}\ . $$ It follows that $\mathbf{k}^2 = -ab\mathbf{1}$ and that any two of $\mathbf{i}, \mathbf{j}, \mathbf{k}$ anti-commute.

The construction is symmetric: $(a,b)_F = (b,a)_F$. The algebra so constructed is a central simple algebra over $F$.

The algebra $(1,1)_F$ is isomorphic to the $2 \times 2$ matrix ring $M_2(F)$. Such a quaternion algebra is termed split.

References

• Tsit-Yuen Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67 , American Mathematical Society (2005) ISBN 0-8218-1095-2 Zbl 1068.11023 MR2104929

Square-free

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 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 over the real or complex numbers, this is equivalent to having no repeated roots.

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.

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 $m$.

In the field of complex numbers, there are roots of unity of every order: those of order $m$ take the form $$ \cos \frac{2\pi k}{m} + i \sin \frac{2\pi k}{m} $$ where $0 < k < m$ and $k$ is relatively prime to $m$.

References

[1]	S. Lang, "Algebra" , Addison-Wesley (1984)

Law of quadratic reciprocity

Gauss reciprocity law

A relation connecting the values of the Legendre symbols (cf. Legendre symbol) $(p/q)$ and $(q/p)$ for different odd prime numbers $p$ and $q$ (cf. Quadratic reciprocity law). In addition to the principal reciprocity law of Gauss for quadratic residues, which may be expressed as the relation

$$\left(\frac pq\right)\left(\frac qp\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

there are two more additions to this law, viz.:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}\quad\text{and}\quad\left(\frac2p\right)=(-1)^{(p^2-1)/8}.$$

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 [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 and biquadratic residues led Gauss to introduce the ring of Gaussian integers.

References

[1]	C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[2]	I.M. [I.M. Vinogradov] Winogradow, "Elemente der Zahlentheorie" , R. Oldenbourg (1956) (Translated from Russian)
[3]	H. Hasse, "Vorlesungen über Zahlentheorie" , Springer (1950)

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.

Quadratic reciprocity law

The relation

$$\left(\frac pq\right)\left(\frac pq\right)=(-1)^{(p-1)/2\cdot(q-1)/2},$$

connecting the Legendre symbols (cf. Legendre symbol)

$$\left(\frac pq\right)\quad\text{and}\quad\left(\frac qp\right)$$

for different odd prime numbers $p$ and $q$. There are two additions to this quadratic reciprocity law, namely:

$$\left(\frac{-1}{p}\right)=(-1)^{(p-1)/2}$$

and

$$\left(\frac 2p\right)=(-1)^{(p^2-1)/8}.$$

C.F. Gauss gave the first complete proof of the quadratic reciprocity law, which for this reason is also called the 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. 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 $p$.

References

[1]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[2]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)

Comments

See also Quadratic residue; Dirichlet character.

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapt. XIII

Euler function

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

$$\sum_{d|n}\phi(d)=n,$$

$$c\frac{n}{\ln\ln n}\leq\phi(n)\leq n,$$

$$\sum_{n\leq x}\phi(n)=\frac{3}{\pi^2}x^2+O(x\ln x).$$

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. [a1].

For a derivation of the asymptotic formula in the article above, as well as of the formula

$$\lim_{n\to\infty}\inf\phi(n)\frac{\ln\ln n}{n}=e^{-\gamma},$$

where $\gamma$ is the Euler constant, see also [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 $x < 10^{1000000}$, [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 [c2]. For some results on this still (1996) largely open problem, see [c3] and the references therein.

References

[1]	K. Chandrasekharan, "Introduction to analytic number theory" , Springer (1968) MR0249348 Zbl 0169.37502
[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8 MR0568909 Zbl 0423.10001
[c1]	A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" Math. Comp. , 63 (1994) pp. 415–419
[c2]	D.H. Lehmer, "On Euler's totient function", Bulletin of the American Mathematical Society 38 (1932) 745–751 DOI 10.1090/s0002-9904-1932-05521-5 Zbl 0005.34302
[c3]	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
[c4]	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
[c5]	R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188
[c6]	R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190
[c7]	L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155

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 functions.

One has $$ J_k(n) = n^k \prod_{p|n} \left({ 1 - p^{-k} }\right) $$ where $p$ runs over the prime numbers dividing $n$, and $$ J_k(n) = \sum_{d | n} \mu(n/d) d^k $$ 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.

References

[a1]	A. Schlafly, S. Wagon, "Carmichael's conjecture on the Euler function is valid below $10^{1000000}$" Math. Comp. , 63 (1994) pp. 415–419
[a2]	D.H. Lehmer, "On Euler's totient function" Bull. Amer. Math. Soc. , 38 (1932) pp. 745–751
[a3]	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
[a4]	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
[a5]	R. Sivamarakrishnan, "The many facets of Euler's totient II: generalizations and analogues" Nieuw Archief Wiskunde (4) , 8 (1990) pp. 169–188
[a6]	R. Sivamarakrishnan, "The many facets of Euler's totient I" Nieuw Archief Wiskunde (4) , 4 (1986) pp. 175–190
[a7]	L.E. Dickson, "History of the theory of numbers I: Divisibility and primality" , Chelsea, reprint (1971) pp. Chapt. V; 113–155

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 $(k + 1)$-tuple together with $n$. This is a generalisation of Euler's totient function, which is $J_1$.

Jordan's totient function is multiplicative and may be evaluated as $$ J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) \ . $$

By Möbius inversion we have $\sum_{d | n } J_k(d) = n^k $. The 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, [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 [c5], [c6].

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 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. Zbl 1079.11001

Matrix multiplication

A binary operation on compatible matrices over a ring $R$. There are several such operations.

Cayley multiplication

Most usually what is referred to as "matrix multiplication". The product of an $m \times n$ matrix $A$ and an $n \times p$ matrix $B$ is the $m \times p$ matrix $AB$ with entries $$ (AB)_{ik} = \sum_{j=1}^n a_{ij} b_{jk}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,p\,. $$

The multiplication corresponds to composition of linear maps. If $A$ is the matrix of a linear map $\alpha : R^m \rightarrow R^n$ and $B$ is the matrix of a linear map $\beta : R^n \rightarrow R^p$, then $AB$ is the matrix of the linear map $\alpha\beta : R^m \rightarrow R^p$.

Hadamard multiplication

The Hadamard product, or Schur product, of two $m \times n$ matrices $A$ and $B$ is the $m \times n$ matrix $AB$ with $$ (A \circ B)_{ij} = a_{ij} b_{ij}\ ,\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,. $$

Kronecker multiplication

The Kronecker product, also tensor product or direct product, of an $m \times n$ matrix $A$ and an $p \times q$ matrix $B$ is the $mp \times nq$ matrix $AB$ with entries $$ (A \otimes B)_{(i-1)p+k,(j-1)q+l} = a_{ij} b_{kl},\ \ i=1,\ldots,m\,\ j=1,\ldots,n\,\ k=1,\ldots,p\,\ l=1,\ldots,q\,. $$

References

• Gene H. Golub, Charles F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences 3, JHU Press (2013) ISBN 1421407949
• James E. Gentle, Matrix Algebra: Theory, Computations, and Applications in Statistics, Springer Texts in Statistics, Springer (2007) ISBN 0-387-70872-3
• Manfred Schroeder, Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, Springer (2008) ISBN 3-540-85297-2

Multiplicative sequence

Also m-sequence, a sequence of polynomials associated with a formal group structure. They have application in the cobordism ring in algebraic topology.

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

$$\sum_i p_i z^i = \sum p'_i z^i \cdot \sum_i p''_i z^i $$

implies

$$\sum_i K_i(p_1,\ldots,p_i) z^i = \sum_j K_j(p'_1,\ldots,p'_j) z^j \cdot \sum_k K_k(p''_1,\ldots,p''_k) z^k . $$ The power series

$$\sum K_n(1,0,\ldots,0) z^n $$

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

$$ \prod_{i=1}^m Q(\beta_i z) $$

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 functions $p$ of the $\beta$. Then $K_j$ defines a multiplicative sequence.

Examples

As an example, the sequence $K_n = p_n$ is multiplicative and has characteristic power series $1+z$.

Consider the power series

$$ Q(z) = \frac{\sqrt z}{\tanh \sqrt z} = 1 - \sum_{k=1}^\infty (-1)^k \frac{2^{2k}}{(2k)!} B_k z^k $$ 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

$$ Q(z) = \frac{2\sqrt z}{\sinh 2\sqrt z} $$

is denoted $A_j(p_1,\ldots,p_j)$.

The multiplicative sequence with characteristic power series

$$Q(z) = \frac{z}{1-\exp(-z)} = 1 + \frac{x}{2} - \sum_{k=1}^\infty (-1)^k \frac{B_k}{(2k)!} z^{2k} $$ is denoted $T_j(p_1,\ldots,p_j)$: the Todd polynomials.

Genus

The genus of a multiplicative sequence is a ring homomorphism, from the cobordism ring of smooth oriented compact manifolds to another ring, usually the ring of rational numbers.

For example, the Todd genus is associated to the Todd polynomials $T_j$ with characteristic power series $$\frac{z}{1-\exp(-z)}$$ and the L-genus is associated to the polynomials $L_j$ with charac\teristic polynomial $$\frac{\sqrt z}{\tanh \sqrt z} . $$

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. Zbl 0843.14009.

Nagao's theorem

A result, named after Hirosi Nagao, about the structure of the group of 2-by-2 invertible matrices over the ring of polynomials over a field. It has been extended by Serre to give a description of the structure of the corresponding matrix group over the coordinate ring of a projective curve.

Nagao's theorem

For a 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

$$ B(R) = \left\lbrace{ \left({\begin{array}{*{20}c} a & b \\ 0 & d \end{array}}\right) : a,d \in R^*, ~ b \in R }\right\rbrace \ . $$

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 $B(K)$.

Serre's extension

In this setting, $C$ is a 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 $\pi_1(G,T)$.

References

• Mason, A.. "Serre's generalization of Nagao's theorem: an elementary approach". Transactions of the American Mathematical Society 353 (2001) 749–767. DOI 10.1090/S0002-9947-00-02707-0 Zbl 0964.20027.
• Nagao, Hirosi. "On $GL(2, K[x])$". J. Inst. Polytechn., Osaka City Univ., Ser. A 10 (1959) 117–121. MR0114866. Zbl 0092.02504.
• Serre, Jean-Pierre. Trees. (Springer, 2003) ISBN 3-540-44237-5.

Erdős–Wintner theorem

A result in probabilistic number theory characterising those additive functions that possess a limiting distribution.

Limiting distribution

A distribution function $F$ is a non-decreasing function from the real numbers to the unit interval [0,1] which is right-continuous and has limits 0 at $-\infty$ and 1 at $+\infty$.

Let $f$ be a complex-valued function on natural numbers. We say that $F$ is a limiting distribution for $f$ if $F$ is a distribution function and the sequence $F_N$ defined by

$$ F_n(t) = \frac{1}{N} | \{n \le N : |f(n)| \le t \} | $$

converges weakly to $F$.

Statement of the theorem

Let $f$ be an additive function. There is a limiting distribution for $f$ if and only if the following three series converge: $$ \sum_{|f(p)|>1} \frac{1}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)}{p} \,,\ \sum_{|f(p)|\le1} \frac{f(p)^2}{p} \ . $$

When these conditions are satisfied, the distribution is given by $$ F(t) = \prod_p \left({1 - \frac{1}{p} }\right) \cdot \left({1 + \sum_{k=1}^\infty p^{-k}\exp(i t f(p)^k) }\right) \ . $$

References

• Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. Handbook of number theory I. Dordrecht: Springer-Verlag (2006). pp. 564–566. ISBN 1-4020-4215-9. Zbl 1151.11300
• Tenenbaum, Gérald Introduction to Analytic and Probabilistic Number Theory. Cambridge studies in advanced mathematics 46. Cambridge University Press (1995). ISBN 0-521-41261-7. Zbl 0831.11001

Brauer–Wall group

A group classifying graded central simple algebras 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 algebras defines the Brauer–Wall group $\mathrm{BW}(F)$.[Lam (2005) pp.98–99]

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 $$ 0 \rightarrow \mathrm{B}(F) \rightarrow \mathrm{BW}(F) \rightarrow Q(F) \rightarrow 0 $$ 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.

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] which has kernel $I^3$, where $I$ is the fundamental ideal of $W(F)$.[Lam (2005) p.115]

Examples

• $\mathrm{BW}(\mathbf{R})$ is isomorphic to $\mathbf{Z}/8$. This is an algebraic aspect of Bott periodicity.

References

• Lam, Tsit-Yuen, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, (American Mathematical Society, 2005) ISBN 0-8218-1095-2 MR2104929, Zbl 1068.11023
• Wall, C. T. C., "Graded Brauer groups", Journal für die reine und angewandte Mathematik 213 (1964) 187–199, ISSN 0075-4102, Zbl 0125.01904, MR0167498

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 $$ c(h,k)^g c(hk,g) = c(h,kg) c(k,g) \ . $$

Cocycles are equivalent if there exists some system of elements $a : G \rightarrow L^*$ with $$ c'(g,h) = c(g,h) (a_g^h a_h a_{gh}^{-1}) \ . $$

Cocycles of the form $$ c(g,h) = a_g^h a_h a_{gh}^{-1} $$ are called split. Cocycles under multiplication modulo split cocycles form a group, the second cohomology group $H^2(G,L^*)$.

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 $$ \lambda u_g = u_g \lambda^g \ , $$ $$ u_g u_h = u_{gh} c(g,h) \ . $$ Equivalent factor systems correspond to a change of basis in $A$ over $K$. We may write $$ A = (L,G,c) \ .$$

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]

Cyclic algebra

Let us further restrict to the case that $L/K$ is 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 $$ u_{t^i} = u^i $$ and then we have $u^n = a$ in $K$. This element $a$ specifies a cocycle $c$ by $$ c(t^i,t^j) = \begin{cases} 1 & \text{if } i+j < n, \\ a & \text{if } i+j \ge n. \end{cases} $$

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.