Central simple algebra

A central simple algebra over a field $k$ is a finite-dimensional associative $k$-algebra that is a simple ring and whose center is exactly $k$.

For a field of characteristic other than two and nonzero $a,b$, the quaternion algebra has basis $1,i,j,ij$ with the displayed relations. It is a four-dimensional central simple algebra. After adjoining a square root of $a$, diagonalizing $i$ and taking $j=\left(\begin{array}{cc}0& b\\ 1& 0\end{array}\right)$ identifies it with $M_2$. Quaternion conjugation fixes scalars and negates $i,j,ij$, giving norm $x_0^2-a{x}_1^2-b{x}_2^2+ab{x}_3^2$. An element is invertible exactly when its norm is nonzero. Thus the quaternion algebra is either a division algebra or a split matrix algebra. The familiar quaternion algebra is $(-1,-1{)}_{\mathbb{R}}$.

For $p\equiv 3(\mathrm{mod}4)$ and $z=x+iy\ne 0$ with rational $x,y$, divide $x^2+y^2$ by the square of the common minimum $p$-adic valuation. Its remaining unit cannot vanish modulo $p$, because $-1$ is not a square there. The norm therefore has even valuation. In particular a rational parameter $a$ with odd valuation at such a prime is not a norm, and $(-1,a{)}_{\mathbb{Q}}$ is a division quaternion algebra: its norm $N(z)-aN(w)$ cannot vanish for a nonzero element $z+wj$.

The reduced norm of an element of a central simple algebra of degree $d$ is its determinant after any splitting isomorphism to $M_d$. Inner automorphisms of a matrix algebra preserve that determinant, while Galois invariance descends its polynomial coefficients to the center. It is a multiplicative homogeneous degree-$d$ polynomial, vanishing exactly on noninvertible elements. On a central division algebra it is anisotropic.

Every $L$-algebra automorphism of $M_d(L)$ is conjugation by an invertible matrix. For matrix units, the vectors $h(E_{i1})w$ with $0\ne w\in \mathrm{im}h(E_{11})$ form a basis in which $h$ acts in the usual way on all matrix units. Scalar conjugating matrices act trivially, giving ${\mathrm{Aut}}_L(M_d(L))={\mathrm{PGL}}_d(L)$.

An extension $E/K$ splits a central simple algebra $A$ of degree $d$ when $A{\otimes }_{K}E\cong M_d(E)$. A finite separable splitting field exists. Splitting is preserved by further scalar extension and is equivalent to the vanishing of the restricted Brauer group class.

A central simple algebra $A$ has dimension $d^2$ over its center. Its degree is this integer $d$. Over a splitting field it becomes $M_d$, so degree records the matrix size independently of the chosen splitting.

Every finite-dimensional semisimple algebra is a finite product of matrix algebras over division algebras. In particular, every central simple algebra over $k$ is isomorphic to $M_r(D)$ for a finite-dimensional central division algebra $D$ over $k$. The integer $r$ and the isomorphism class of $D$ are unique.

Indeed, decompose the right regular module of a finite-dimensional semisimple algebra $A$ aswhere the $S_i$ are pairwise nonisomorphic simple modules. The Schur lemma makes $D_i={\mathrm{End}}_A(S_i)$ a division ring and gives ${\mathrm{Hom}}_A(S_i,S_j)=0$ for $i\ne j$. Left multiplication identifiesConversely, the column module over $M_n(D)$ is simple and the regular module is a direct sum of $n$ copies of it, so finite products of such matrix rings are semisimple. The simple-module decomposition also proves uniqueness up to permuting the factors.

For a nonzero finite-dimensional complex vector space $V$, a unital matrix algebra acting irreducibly on $V$ is the full endomorphism algebra. Its Jacobson radical annihilates $V$: otherwise the radical times $V$ is all of $V$, contradicting nilpotence. Faithfulness kills the radical, and the Artin–Wedderburn theorem then leaves one simple matrix factor acting on its standard column module. This is a matrix-algebra theorem, distinct from the orbit-counting Burnside lemma and the finite-group Burnside's theorem.

The tensor product over $k$ of two central simple $k$-algebras is central simple. After extending scalars to an algebraic closure, both factors and their tensor product become full matrix algebras; faithful flatness then descends simplicity and the center.

The Brauer group of a field $k$ consists of Morita equivalence classes of central simple $k$-algebras. Multiplication is induced by tensor product, the identity is $[k]$, and the inverse of $[A]$ is the class of the opposite algebra $A^{\mathrm{op}}$.

The Brauer class of a central simple algebra is its equivalence class under adjoining matrix-algebra factors. Two algebras have the same class exactly when their central division-algebra representatives are isomorphic. The zero class consists of split matrix algebras; the inverse of $[A]$ is the class of the opposite algebra. For a cyclic algebra $(L/K,\sigma ,a)$, the class is zero exactly when $a$ is a field norm from $L$.

For a number field $K$, localization of the Brauer group is injective. Its image is exactly the finite-support tuples of local classes whose local Brauer invariants sum to zero. The non-Archimedean summands have invariant group $\mathbb{Q}/\mathbb{Z}$, real summands have $\{0,1/2\}$, and complex summands are zero. For a cyclic character $\chi$ and parameter $a$, the invariant of the local cyclic algebra is $\chi ({\mathrm{Art}}_{K_v}(a))$. Thus the zero-sum condition on principal parameters is the additive form of Artin reciprocity.

Associate to each prime $p\equiv 3(\mathrm{mod}4)$ the class of $(-1,p{)}_{\mathbb{Q}}$. These classes have order two, since quaternion conjugation identifies the algebra with its opposite. The product rule $[(-1,a)][(-1,b)]=[(-1,ab)]$ follows by factoring their tensor product into the commuting subalgebras generated by $i_1,j_1{j}_2$ and by $i_1{i}_2,j_2$; the second is the split algebra $(1,b)$. Every nonempty finite product of distinct such primes has an odd valuation at one of them. The quaternion norm obstruction at primes congruent to three modulo four therefore makes its class nonzero, proving independence. Euclid's construction $4p_1\cdots p_m-1$ proves there are infinitely many such primes.

In characteristic $p$, the exact sequence $1\to K_s^{\times }\stackrel{p}{\to }{K}_s^{\times }\to K_s^{\times }/(K_s^{\times }{)}^p\to 1$ has a quotient killed by $p$. The characteristic p Galois dimension bound makes its second cohomology zero. Hence multiplication by $p$ on $\mathrm{Br}(K)$ is surjective. The root map is injective, and its failure of surjectivity on a separable closure is encoded by the quotient.

For a complete discretely valued field $K$ with perfect residue field $k$, there is a split exact sequence $0\to \mathrm{Br}(k)\to \mathrm{Br}(K)\stackrel{{\partial }_K}{\to }{H}^1(k,\mathbb{Q}/\mathbb{Z})\to 0$. The residue is canonical. A uniformizer chooses a splitting $\chi \mapsto (\chi ,\pi )$; this choice does not make the residue itself noncanonical.

For a local field with finite residue field, evaluate the Witt residue character on arithmetic Frobenius. This gives ${\mathrm{inv}}_K:\mathrm{Br}(K)\cong \mathbb{Q}/\mathbb{Z}$. For the unramified degree-$m$ cyclic algebra with arithmetic Frobenius generator and parameter $a$, the invariant is $v_K(a)/m$. Its reduced denominator is the index of the division representative.

For a finite Galois extension $L/K$ of non-Archimedean local fields, its relative Brauer group is the kernel of restriction $\mathrm{Br}(K)\to \mathrm{Br}(L)$ and identifies with $H^2(\mathrm{Gal}(L/K),L^{\times })$. Restriction multiplies the local Brauer invariant by $[L:K]$. This kernel is therefore cyclic of order $[L:K]$, and the indicated class is its distinguished generator. In an unramified extension with arithmetic Frobenius generator it is represented by the cyclic algebra with parameter a uniformizer.

For a finite extension of local fields $L/K$, restriction multiplies the local Brauer invariant by $[L:K]$ and corestriction preserves it. For restriction, the residue-character value multiplies by the residue degree and the parameter valuation by the ramification index. For corestriction, lift the residue character using divisibility of $\mathbb{Q}/\mathbb{Z}$ and use the norm projection formula for cyclic Brauer pairings.

The period is the order of a class in the Brauer group. It divides its index. Over a local field with finite residue field, the period and index are both the reduced denominator of its local Brauer invariant.

If $A\cong M_r(D)$ with $D$ a central division algebra, its index is $\mathrm{deg}D$. It depends only on the Brauer group class. The index divides every finite splitting-field degree and equals the degree of a maximal subfield of the division representative.

For a finite Galois extension $L/k$, Galois group $G$, and normalized two-cocycle $\varphi :G\times G\to L^{\times }$, the crossed-product algebra iswith $u_{\sigma }a=\sigma (a)u_{\sigma }$ and $u_{\sigma }{u}_{\tau }=\varphi (\sigma ,\tau )u_{\sigma \tau }$. It is a central simple $k$-algebra split by $L$.

For a cyclic extension $E/K$ of degree $m$ with generator $\sigma$ and $a\in K^{\times }$, the cyclic algebra is generated by $E$ and $u$ with $ub=\sigma (b)u$, $u^m=a$. It is a central simple algebra of degree $m$. Its Brauer class can also be written $(\chi ,a)$, where $\chi (\sigma )=1/m$.

If $a=N_{L/K}c$, let $L$ act on itself by multiplication and let the extra generator act by $c\sigma$. Its $n$th power is multiplication by $a$, and these operators give the cyclic algebra as ${\mathrm{End}}_K(L)$. Conversely, if the algebra is a matrix algebra on an $n$-dimensional vector space, its embedded degree-$n$ field $L$ makes that space one-dimensional over $L$. The extra generator is then $c\sigma$ in an $L$-basis, so its $n$th power forces $a=Nc$.

For a finite extension $L/K$, an unramified or general cyclic character $\chi$ over $K$, and $a\in L^{\times }$, transfer satisfies ${\mathrm{Cor}}_{L/K}(\mathrm{Res}\chi ,a)=(\chi ,N_{L/K}a)$. It is the cup-product projection formula combined with the field norm on multiplicative degree-zero coefficients. With the norm definition of Brauer transfer it also covers purely inseparable steps.

Cohomologous normalized two-cocycles define isomorphic crossed-product algebras: replacing $\varphi$ by the coboundary associated with a one-cochain rescales the basis elements $u_{\sigma }$. Consequently the crossed-product construction induces a map