Central Simple Algebras and the Brauer Group

Introduction

A central simple algebra over a field $F$ is a finite-dimensional $F$-algebra with centre exactly $F$ and no two-sided ideal other than $0$ and the algebra itself. These are the algebras that behave, over an arbitrary field, as the matrix algebras behave over an algebraically closed one: every one of them becomes a full matrix algebra after a suitable extension of scalars, and every one of them is a matrix algebra over a division algebra, by Wedderburn's structure theorem. They are the natural home of the Skolem–Noether theorem, of the notion of similarity, and of the tensor product; and the classes of central simple algebras under similarity form a group, the Brauer group $\operatorname{Br}(F)$, which is the subject of this article.

The article develops the theory from the definition of a central simple algebra. The facts used are stated as standard: that a finite-dimensional algebra over a field without zero divisors is a division algebra; that the tensor product of central simple algebras is central simple; Frobenius' theorem; Wedderburn's little theorem; and the Skolem–Noether theorem. What this article adds is the systematic theory: the structure and dimension of a central simple algebra, the index and the exponent, splitting fields, the crossed-product description of the classes, the $2$-torsion and its quaternion generators, and the computation of the Brauer group for the standard fields. The finer identification of the $2$-torsion with the quadratic forms by the Clifford invariant needs the forms of Part II and is named only. The generalisation from a field to a commutative ring is not developed here.

Everything is algebraic. The finer arithmetic of the Brauer group and its cohomological interpretation of the relative Brauer group over a local or global field are named where they belong and deferred to the articles that own them; in particular, the class-field-theoretic identifications belong to the Part I article Class Field Theory, in the category Rings and Fields.

Central Simple Algebras

Definition and first properties

Definition. Let $F$ be a field. A finite-dimensional $F$-algebra $A$ with $1 \neq 0$ is central if its centre is $F \cdot 1_A$, and central simple if in addition the only two-sided ideals of $A$ are $0$ and $A$.

Throughout this article $F$ is a field and $A$, $B$ are finite-dimensional unital associative $F$-algebras, and all tensor products are over $F$ unless a subscript says otherwise.

Proposition. Let $A$ be central simple over $F$.

$A$ is a division algebra if and only if $A$ has no zero divisors, by the standard finite-dimensional criterion (a finite-dimensional algebra over a field with no zero divisors is a division algebra).
Every nonzero two-sided ideal of $A \otimes_F L$ meets $A$ trivially, for every field extension $L/F$; equivalently $A \otimes_F L$ is simple.
The centre of $A \otimes_F L$ is $L \cdot 1$, so $A \otimes_F L$ is central simple over $L$.

Proof. Statement 1 is the standard finite-dimensional criterion. For 2 and 3, the centre statement is standard: it is proved by extending scalars to an algebraic closure, where the algebra becomes a matrix algebra; the simplicity is the standard argument that a nonzero ideal of $A \otimes_F \bar F$ intersects $M_n(\bar F)$ in a nonzero ideal, hence contains a matrix unit, and the matrix units generate.

Thus centrality and simplicity are preserved by base change, which is the technical heart of the theory: an $F$-algebra is central simple exactly when it becomes a matrix algebra over a suitable extension, in a sense made precise by the notion of a splitting field below.

Examples

(a) Matrix algebras. For every $n \geq 1$, the algebra $M_n(F)$ is central simple: its centre is $F I_n$, by the computation, and it is simple because the matrix units generate it from any nonzero element.

(b) Division algebras. A finite-dimensional division algebra $D$ over $F$ is simple, and it is central simple exactly when its centre is $F$.

(c) Quaternion algebras. Let $F$ be a field of characteristic not $2$ and let $a, b \in F^\times$. The quaternion algebra $(a,b)_F$ has $F$-basis $1, u, v, w$ with

$$ u^2 = a, \qquad v^2 = b, \qquad uv = w = -vu . $$

It is central: $u, v, w$ each anticommute with one another and the only central elements are the scalars. It is simple of dimension $4$, hence, by Wedderburn's structure theorem below, either a division algebra or $M_2(F)$. For $F = \mathbb{R}$ and $a = b = -1$ it is $\mathbb{H}$; for $a = b = 1$ the element $1 + u$ is a zero divisor and the algebra is $M_2(\mathbb{R})$. The quaternion algebras are the four-dimensional central simple algebras, and their classification is the first case of the general theory.

(d) The biquaternions are not central over $\mathbb{R}$. The algebra $\mathbb{B} = \mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}$ has centre $\mathbb{C}$, so it is simple but not central over $\mathbb{R}$; over $\mathbb{C}$ it is $M_2(\mathbb{C})$ and is central simple. This is the distinction that the phrase "central over $F$" records, and it is why the ground field is named at each step.

(e) A non-example. The split-complex algebra $\mathbb{D} = \mathbb{R}[x]/(x^2-1)$ is commutative and the dual numbers $\mathbb{D}' = \mathbb{R}[x]/(x^2)$ are not simple. Neither is central simple. The group algebra $\mathbb{C}[G]$ of a finite group is central simple only in the trivial case $G = 1$: otherwise it is a product of two or more ideals.

The dimension is a square

Theorem. Let $A$ be central simple over $F$ of finite dimension $n = \dim_F A$. Then $A \otimes_F \bar F \cong M_d(\bar F)$ where $\bar F$ is an algebraic closure of $F$, and $n = d^2$.

Proof. By the proposition above, $A \otimes_F \bar F$ is central simple over the algebraically closed field $\bar F$. Over an algebraically closed field the only finite-dimensional division algebra is the field itself; and Wedderburn's structure theorem writes a central simple algebra over $\bar F$ as $M_d(\bar F)$. Comparing dimensions, $n = d^2$.

The integer $d$ is the degree of $A$, written $\deg(A)$. It is a numerical invariant of $A$ and not of a presentation: the dimension of a central simple algebra is always a perfect square. The theorem also shows that simplicity is a property detectable after base change, since the matrix algebra $M_d(\bar F)$ is visibly central simple.

The Structure of Central Simple Algebras

Wedderburn's structure theorem

Theorem (Wedderburn, standard). Every finite-dimensional central simple $F$-algebra $A$ is isomorphic to a matrix algebra $M_n(D)$ over a central $F$-division algebra $D$, with $n \geq 1$ and $D$ determined up to isomorphism. Equivalently, writing $A = \operatorname{End}_D(V)$ for the unique simple $A$-module $V = D^n$, the division algebra is $D = \operatorname{End}_A(V)^{\mathrm{op}}$.

Pro. The algebra $A$ is simple and finite-dimensional, so by the Wedderburn–Artin theorem it is $M_n(D)$ for a division algebra $D$; the centre of $M_n(D)$ is the centre of $D$, computed, so centrality of $A$ makes $D$ central over $F$. Uniqueness follows because $D$ is recovered as $\operatorname{End}_A(V)^{\mathrm{op}}$ for the unique simple module $V$, and the simple module is unique because $A$ is simple.

The theorem reduces the classification of central simple algebras to the classification of central division algebras together with the integer $n$. The division algebra $D$ is called the division algebra part of $A$, and

$$ \deg(A) = n \sqrt{\dim_F D}, \qquad \operatorname{ind}(A) := \sqrt{\dim_F D} $$

defines the index of $A$. The index is a positive integer, $\deg(A) = n \cdot \operatorname{ind}(A)$, and $\operatorname{ind}(A) = 1$ exactly when $A$ is a matrix algebra over $F$, in which case $A$ is called split.

Skolem–Noether and inner automorphisms

Theorem (Skolem–Noether, standard). Let $A$ be central simple over $F$, let $B$ be a simple $F$-subalgebra of $A$, and let $f, g : B \to A$ be two unital $F$-algebra homomorphisms. Then there is a unit $a \in A^\times$ with

$$ g(b) = a\, f(b)\, a^{-1} \qquad \text{for all } b \in B . $$

The theorem is stated as standard, and two corollaries are used repeatedly.

Corollary. For a central simple algebra $A$ the map $A^\times \to \operatorname{Aut}_F(A)$, $a \mapsto (x \mapsto axa^{-1})$, is surjective with kernel $F^\times$, so

$$ \operatorname{Aut}_F(A) \cong A^\times / F^\times . $$

In particular two matrix algebras $M_n(D)$ and $M_m(D')$ over division algebras are isomorphic if and only if $n = m$ and $D \cong D'$, and the embeddings of a separable subfield of $A$ into $A$ are conjugate.

Example (the real quaternions, and the commutative boundary). For $A = \mathbb{H}$ over $\mathbb{R}$ the corollary gives $\operatorname{Aut}_{\mathbb{R}}(\mathbb{H}) \cong \mathbb{H}^\times/\mathbb{R}^\times$: every $\mathbb{R}$-automorphism of $\mathbb{H}$ is conjugation by a unit, the kernel of $\mathbb{H}^\times \to \operatorname{Inn}(\mathbb{H})$ being the centre $\mathbb{R}^\times$. The same computation is carried out in Division Algebras. In the commutative case the statement is empty, because $\operatorname{Inn}(A)$ is trivial there; for example $\mathbb{C}$ is simple over $\mathbb{R}$ but not central, and complex conjugation is an $\mathbb{R}$-automorphism of $\mathbb{C}$ that is not inner. Centrality is exactly what the theorem needs and cannot be dropped.

Corollary (the double centralizer theorem). Let $B$ be a simple $F$-subalgebra of $A$ with centre $L$. Then the centralizer $C_A(B)$ is simple, $B \otimes_L C_A(B) \cong A$ as $L$-algebras, and

$$ \dim_F A = \dim_F B \cdot \dim_F C_A(B) . $$

Proof. The centralizer is an $L$-algebra containing $L$, and the map $B \otimes_L C_A(B) \to A$, $b \otimes c \mapsto bc$, is an injective homomorphism of $L$-algebras; since $A$ is simple and $B \otimes_L C_A(B)$ is a tensor product of central simple $L$-algebras by the tensor product theorem below, the map is an isomorphism. Taking dimensions gives the formula.

The reduced trace and the opposite algebra

For a central simple algebra $A$ of degree $d$, choose a splitting field $L$ with $A \otimes_F L \cong M_d(L)$ and define the reduced trace and reduced determinant

$$ \operatorname{Trd} : A \to F, \qquad \operatorname{Nrd} : A \to F $$

so that after base change they become the matrix trace and determinant. Both take values in $F$ and are independent of the splitting field chosen; the reduced trace is $F$-linear, $\operatorname{Trd}(ab) = \operatorname{Trd}(ba)$, and the reduced determinant is multiplicative, $\operatorname{Nrd}(ab) = \operatorname{Nrd}(a)\operatorname{Nrd}(b)$. The opposite algebra $A^{\mathrm{op}}$ has the same underlying $F$-vector space and product $a \cdot_{\mathrm{op}} b = ba$. It is central simple, of the same degree as $A$, and it is anti-isomorphic to $A$; the transpose map on a matrix algebra identifies $M_n(F)^{\mathrm{op}} \cong M_n(F)$, and quaternion conjugation identifies $\mathbb{H}^{\mathrm{op}} \cong \mathbb{H}$.

The opposite algebra is the reason the Brauer group has inverses. For every central simple $A$ of degree $d$ there is an isomorphism

$$ A \otimes_F A^{\mathrm{op}} \;\cong\; \operatorname{End}_F(A) \;\cong\; M_{d^2}(F), $$

the first map being $a \otimes b \mapsto (x \mapsto axb)$, which is an injective algebra homomorphism because the tensor product is simple, and hence an isomorphism by the dimension count $d^2 \cdot d^2 = (d^2)^2$. So the tensor product of a central simple algebra with its opposite is split.

The Brauer Group

The tensor product of central simple algebras

Theorem. If $A$ and $B$ are central simple over $F$, then $A \otimes_F B$ is central simple over $F$, of degree $\deg(A)\deg(B)$ and dimension $(\dim_F A)(\dim_F B)$.

The theorem is standard, and the article uses it as such: the centre is computed from the centre of each factor, and simplicity follows by extending scalars to a splitting field, where the tensor product of two matrix algebras is a matrix algebra. The theorem is what allows the tensor product to be used as a group operation; without it the classes below would not be closed under multiplication.

Similarity

Definition. Two central simple $F$-algebras $A$ and $B$ are similar, written $A \sim B$, if

$$ A \otimes_F M_m(F) \cong B \otimes_F M_n(F) \qquad \text{for some } m, n \geq 1 . $$

By Wedderburn's structure theorem this is equivalent to the assertion that the division algebras $D_A$ and $D_B$ underlying $A$ and $B$ are isomorphic, so similarity is an equivalence relation, and each class has a unique representative up to isomorphism that is a division algebra. We write $[A]$ for the class of $A$ and speak of the division algebra representative of $[A]$.

Lemma. Similarity is compatible with the tensor product: if $A \sim A'$ and $B \sim B'$ then $A \otimes_F B \sim A' \otimes_F B'$.

Proof. It suffices to check that $A \otimes M_m(F) \sim A$ for all $m$, since then

$$ (A \otimes M_m(F)) \otimes (B \otimes M_n(F)) \;\cong\; (A \otimes B) \otimes (M_m(F) \otimes M_n(F)) \;\cong\; (A\otimes B)\otimes M_{mn}(F) $$

and the tensor products in the statement may be replaced one factor at a time. But $A \otimes_F M_m(F) \cong M_m(A)$, which is a matrix algebra over $A$, hence similar to $A$ by definition.

The group law

Theorem (the Brauer group). The set of similarity classes of central simple $F$-algebras, with product

$$ [A] \cdot [B] := [A \otimes_F B] $$

and identity the class of $F$, is an abelian group, the Brauer group $\operatorname{Br}(F)$. The inverse of $[A]$ is $[A^{\mathrm{op}}]$, and every element has finite order.

Proof. The product is well defined by the lemma, associative and commutative because the tensor product of algebras is associative and commutative up to canonical isomorphism, and unital because $A \otimes_F F \cong A$. The class of $A^{\mathrm{op}}$ is inverse to that of $A$ by the isomorphism $A \otimes_F A^{\mathrm{op}} \cong M_{d^2}(F)$ above. Finiteness of the order is the theorem of the next subsection: the order of $[A]$ is the exponent, and it divides the index, which is finite.

The group is abelian but generally not finite: over $\mathbb{Q}$ and over every number field it is infinite, by the theorem of Hasse, Brauer and Noether recalled below.

Splitting fields, index and exponent

Definition. A splitting field of a central simple $F$-algebra $A$ is a field extension $L/F$ with

$$ A \otimes_F L \;\cong\; M_d(L), \qquad d = \deg(A) . $$

Every central simple algebra has a splitting field: an algebraic closure of $F$ splits it, by the dimension theorem of the second section. The interest of the notion is that a splitting field of finite degree exists, and the degree of the smallest one is the index.

Theorem (standard). Let $A$ be central simple over $F$, of degree $d$ and index $e$. Then:

A field extension $L/F$ splits $A$ if and only if $L$ contains a subfield isomorphic to a maximal subfield of the division algebra representative $D$ of $A$.
The index $e$ is the degree of every maximal subfield of $D$, and every such subfield has degree $e$ over $F$.
There is a splitting field of degree $e$ over $F$, namely any maximal subfield of $D$; consequently the index divides the degree of every splitting field, and in particular $e \mid d = \deg(A)$.
The exponent $\exp(A)$, the order of $[A]$ in $\operatorname{Br}(F)$, divides the index $e$.

Proof. Statements 1 and 2 are the standard theory of maximal subfields of a division algebra; the dimension count $\dim_F D = e^2$ and the double centralizer theorem give that a maximal subfield has degree $e$. For 3, a maximal subfield $L$ of $D$ has $D \otimes_F L \cong M_e(L)$ because $D$ becomes split over its own maximal subfield, the centralizer of $L$ in $D$ being $L$ itself; and every splitting field has degree divisible by $e$ by the same argument applied to $D \otimes_F L$. For 4, the exponent divides the index: if $K$ is a maximal subfield of $D$, then $[D]$ lies in $\operatorname{Br}(K/F)$, and the restriction–corestriction identity of the cohomological theory gives $[D]^{[K:F]} = [D]^{e} = 0$ in $\operatorname{Br}(F)$, since restriction to $K$ kills the class; alternatively the same conclusion follows from the reduced determinant and the theory of the reduced characteristic polynomial. Either argument is the standard one, and both are recorded in the references.

The two invariants are related by the theorem: every central simple algebra is a crossed product with respect to a splitting field, and the resulting cohomological description of the Brauer group makes the exponent the order of a cohomology class.

The relative Brauer group

For a field extension $L/F$, base change $A \mapsto A \otimes_F L$ is a homomorphism of groups,

$$ \operatorname{Br}(F) \longrightarrow \operatorname{Br}(L), \qquad [A] \longmapsto [A \otimes_F L], $$

well defined by the compatibility of the tensor product with base change and the preservation of centrality and simplicity. Its kernel,

$$ \operatorname{Br}(L/F) := \ker\bigl(\operatorname{Br}(F) \to \operatorname{Br}(L)\bigr), $$

is the relative Brauer group of $L/F$: the classes split by $L$. For a finite Galois extension $L/F$ with group $G$, the crossed-product description identifies

$$ \operatorname{Br}(L/F) \;\cong\; H^2(G, L^\times), $$

the second cohomology group of $G$ with coefficients in the multiplicative group of $L$; the group cohomology is that of Group Cohomology. This is the cohomological face of the theory and the reason the exponent of a class is the order of a cohomology class.

The Two-Torsion and the Quaternion Algebras

The $2$-torsion of the Brauer group is the subgroup

$$ \operatorname{Br}_2(F) = \{[A] \in \operatorname{Br}(F) : 2[A] = 0\} $$

of the classes whose exponent divides $2$. By the crossed-product description these are the classes of the quaternion algebras and of their tensor products.

Remark. The finer identification of this subgroup with the quadratic form theory — the Clifford invariant, the Witt group, the Pfister forms, and the theorem of Merkurjev that $\operatorname{Br}_2(F) \cong I_q^2/I_q^3$ — needs the forms of Part II, is treated in Hilbert Algebras and in The Witt Group and the Grothendieck–Witt Ring, and is named here only. What the present Part supplies is the algebra: the quaternion algebras are the four-dimensional central simple algebras, and their classes generate the $2$-torsion.

Crossed Products and Cyclic Algebras

Cyclic algebras

The most explicit central simple algebras are the cyclic algebras. Let $L/F$ be a cyclic extension of degree $n$ with Galois group generated by $\sigma$, and let $a \in F^\times$. The cyclic algebra $(\chi, a)$ is the $F$-algebra

$$ (\chi, a) = L \oplus Lz \oplus \cdots \oplus Lz^{n-1}, \qquad z^n = a, \qquad z\ell = \sigma(\ell) z \quad (\ell \in L). $$

It is central simple of degree $n$, and its class lies in $\operatorname{Br}(L/F)$. The construction is a special case of the crossed product, with factor set determined by the class of $a$ modulo the subgroup $\operatorname{N}_{L/F}(L^\times)$; the isomorphism classes of such algebras are parametrised by the quotient $F^\times/\operatorname{N}_{L/F}(L^\times)$, and the cyclic algebra is split exactly when $a$ lies in that subgroup. In the smallest case $n = 2$, with $L = F(\sqrt a)$ and $\sigma$ the nontrivial automorphism, the cyclic algebra $(\chi, b)$ is the quaternion algebra

$$ (a, b)_F = F(\sqrt a) \oplus F(\sqrt a) z, \qquad z^2 = b, \qquad z \sqrt a = -\sqrt a\, z , $$

which is the doubled description of the four-dimensional algebra of the first section. The algebra is a division algebra when $b$ is not in the image of $\operatorname{N}_{F(\sqrt a)/F}$ and $M_2(F)$ otherwise.

The crossed-product description

Theorem (Noether–Deuring, standard). Let $L/F$ be a finite Galois extension with group $G$, and let $A$ be a central simple $F$-algebra split by $L$. Then there is a $G$-graded $L$-algebra structure $A \otimes_F L = \bigoplus_{\sigma \in G} A_\sigma$ and a factor set $c : G \times G \to L^\times$ with

$$ A \otimes_F L \;\cong\; \bigoplus_{\sigma \in G} L u_\sigma, \qquad u_\sigma \ell = \sigma(\ell) u_\sigma, \qquad u_\sigma u_\tau = c(\sigma, \tau) u_{\sigma\tau}, $$

and the class of $A$ in $\operatorname{Br}(L/F) \cong H^2(G, L^\times)$ is the cohomology class of the factor set $c$. The factor set is a coboundary exactly when $A$ is split by $F$.

The theorem is the content, where the crossed product $L \rtimes_c G$ is defined and the isomorphism with $H^2(G, L^\times)$ is established; it is cited here because it explains both the name "Brauer group" and the structure of the cyclic algebras above. A central simple algebra need not be a crossed product with respect to every splitting field, and need not be cyclic; the question of which algebras are cyclic is a genuine restriction, settled for local fields by the theorem below and open in general.

Computations of the Brauer Group

Algebraically closed fields and finite fields

Theorem. If $F$ is algebraically closed, then $\operatorname{Br}(F) = 0$.

Proof. Over an algebraically closed field the only finite-dimensional division algebra is $F$ itself, so every central simple algebra is $M_n(F)$ and is similar to $F$.

Theorem (Wedderburn). If $F$ is a finite field, then $\operatorname{Br}(F) = 0$.

Proof. By Wedderburn's little theorem every finite division ring is a field, so the only finite-dimensional central division algebra over $\mathbb{F}_q$ is $\mathbb{F}_q$ itself. Wedderburn's structure theorem then makes every central simple $\mathbb{F}_q$-algebra a matrix algebra over $\mathbb{F}_q$, hence split.

The two results say that the Brauer group measures how far a field is from being algebraically closed or finite, and that the interesting cases are the number fields, the local fields and the function fields.

The real and complex numbers

Over $\mathbb{R}$ the division algebras are $\mathbb{R}, \mathbb{C}, \mathbb{H}$ by Frobenius' theorem, and $\mathbb{C}$ is not central, so the only central division algebra besides $\mathbb{R}$ is $\mathbb{H}$. Therefore

$$ \operatorname{Br}(\mathbb{R}) \;\cong\; \mathbb{Z}/2\mathbb{Z}, $$

generated by $[\mathbb{H}]$, with $[\mathbb{H}]^2 = [\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H}] = [M_4(\mathbb{R})] = 0$; consistently, $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \cong M_4(\mathbb{R})$. Over $\mathbb{C}$, which is algebraically closed, the group is trivial. The relative Brauer group of $\mathbb{C}/\mathbb{R}$ is all of $\operatorname{Br}(\mathbb{R})$, since complexification sends $\mathbb{H}$ to $M_2(\mathbb{C})$; and $\operatorname{Br}(\mathbb{C}/\mathbb{R}) \cong H^2(\mathbb{Z}/2, \mathbb{C}^\times) \cong \mathbb{Z}/2$, the generator being the class of the nontrivial cocycle $\sigma \mapsto -1$ in the sense of the crossed-product theorem.

Number fields and local fields

Theorem (Hasse–Brauer–Noether, standard). Let $F$ be a number field with places $v$, and for a finite-dimensional central division algebra $D$ over $F$ let $\operatorname{inv}_v(D) \in \mathbb{Q}/\mathbb{Z}$ be its Hasse invariant at $v$, which is zero for all but finitely many $v$. Then the map

$$ \operatorname{Br}(F) \longrightarrow \bigoplus_v \mathbb{Q}/\mathbb{Z}, \qquad [D] \longmapsto (\operatorname{inv}_v(D))_v $$

is an isomorphism onto the subgroup of the direct sum consisting of the families whose sum is $0$ in $\mathbb{Q}/\mathbb{Z}$, and the index of $[D]$ is the order of the corresponding element of $\mathbb{Q}/\mathbb{Z}$ at any place with nonzero invariant.

This is the reciprocity law of class field theory, and it belongs to the article Class Field Theory; it is recorded here because it computes the Brauer group of a number field completely and shows that it is infinite, with $\operatorname{Br}(\mathbb{Q}) \cong \bigoplus_p \mathbb{Q}/\mathbb{Z} / \text{(the global relation)}$. For a non-Archimedean local field $F$ the same construction gives $\operatorname{Br}(F) \cong \mathbb{Q}/\mathbb{Z}$, the invariant of a cyclic algebra being $1/n$ for a degree-$n$ unramified cyclic algebra, and for $\mathbb{R}$ it gives $\operatorname{Br}(\mathbb{R}) \cong \frac{1}{2}\mathbb{Z}/\mathbb{Z} \subseteq \mathbb{Q}/\mathbb{Z}$. The local invariant map is additive, so tensor products correspond to sums of invariants, and the global isomorphism is the statement that a central simple algebra over a number field is determined by its local invariants subject to the single reciprocity relation.

A table

Field $F$	$\operatorname{Br}(F)$	Generators and remarks
algebraically closed	$0$	every central simple algebra is $M_n(F)$
finite $\mathbb{F}_q$	$0$	Wedderburn's little theorem
$\mathbb{R}$	$\mathbb{Z}/2\mathbb{Z}$	generated by $[\mathbb{H}]$, of index $2$
$\mathbb{C}$	$0$	algebraically closed
local (non-Archimedean)	$\mathbb{Q}/\mathbb{Z}$	invariant $1/n$ for a degree-$n$ unramified cyclic algebra
number field	$\bigoplus_v \mathbb{Q}/\mathbb{Z}$ with reciprocity	Hasse–Brauer–Noether

Summary

A central simple algebra over a field $F$ is a finite-dimensional $F$-algebra with centre $F$ and no nontrivial two-sided ideal. Its dimension is a perfect square $d^2$, it is a matrix algebra $M_n(D)$ over a central $F$-division algebra $D$ by Wedderburn's structure theorem, and its index $\operatorname{ind}(A) = \sqrt{\dim_F D}$ divides its degree $d$ and equals the degree of every maximal subfield of $D$. The Skolem–Noether theorem makes every $F$-algebra automorphism of a central simple algebra inner, so $\operatorname{Aut}_F(A) \cong A^\times/F^\times$, and the double centralizer theorem computes the centralizer of a simple subalgebra. The tensor product of two central simple algebras is central simple, so the similarity classes of central simple algebras form an abelian group under the tensor product, the Brauer group $\operatorname{Br}(F)$, with identity the class of $F$ and inverse the class of the opposite algebra, since $A \otimes_F A^{\mathrm{op}} \cong M_{d^2}(F)$.

A splitting field of $A$ is a field extension $L/F$ with $A \otimes_F L \cong M_d(L)$; a splitting field of degree $\operatorname{ind}(A)$ always exists, and the exponent of $A$, its order in $\operatorname{Br}(F)$, divides its index. For a finite Galois extension $L/F$ with group $G$, every central simple algebra split by $L$ is a crossed product, and the relative Brauer group is $\operatorname{Br}(L/F) \cong H^2(G, L^\times)$, so the Brauer group is the cohomological home of the factor-set classification; the cyclic algebras $(L/F, \sigma, a)$ are the explicit case. The computations are $\operatorname{Br}(F) = 0$ for algebraically closed and for finite $F$, $\operatorname{Br}(\mathbb{R}) \cong \mathbb{Z}/2$ generated by $[\mathbb{H}]$ with $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \cong M_4(\mathbb{R})$, $\operatorname{Br}(F) \cong \mathbb{Q}/\mathbb{Z}$ for a non-Archimedean local field, and for a number field the Hasse–Brauer–Noether isomorphism onto the families of local invariants with zero sum, which belongs to Class Field Theory.

The $2$-torsion of the Brauer group is generated by the quaternion algebras: the classes of exponent dividing $2$ are the classes of the quaternion algebras and of their tensor products, and the quaternion algebras are the four-dimensional central simple algebras. The finer Clifford invariant, which identifies this subgroup with the Witt ring of quadratic forms and gives $\operatorname{Br}_2(F) \cong I_q^2/I_q^3$ (Merkurjev in characteristic not $2$, Sah in characteristic $2$), needs the forms of Part II and is named only.

Summary of Notation

Symbol	Meaning
$F$, $L$	fields; $L/F$ a field extension
$A$, $B$	central simple $F$-algebras
$A^{\mathrm{op}}$	opposite algebra, $a \cdot_{\mathrm{op}} b = ba$
$M_n(F)$	matrix algebra, central simple of dimension $n^2$
$D$	central $F$-division algebra, the division algebra part of $A$
$(a,b)_F$	quaternion algebra, $u^2 = a$, $v^2 = b$, $uv = -vu$
$\mathbb{H}, \mathbb{B}$	real quaternions; biquaternions, central over $\mathbb{C}$ but not over $\mathbb{R}$
$\deg(A) = \sqrt{\dim_F A}$	degree of a central simple algebra
$\operatorname{ind}(A) = \sqrt{\dim_F D}$	index, the degree of a maximal subfield of $D$
$\exp(A)$	exponent, the order of $[A]$ in $\operatorname{Br}(F)$
$A \sim B$	similarity: $A \otimes M_m(F) \cong B \otimes M_n(F)$ for some $m,n$
$\operatorname{Br}(F)$	Brauer group: similarity classes under $\otimes_F$
$\operatorname{Br}(L/F)$	relative Brauer group, classes split by $L$
$\operatorname{Br}_2(F)$	$2$-torsion, classes of exponent dividing $2$
$\operatorname{Trd}, \operatorname{Nrd}$	reduced trace and reduced determinant
$C_A(B)$	centralizer of $B$ in $A$
$\operatorname{Aut}_F(A) \cong A^\times/F^\times$	automorphisms, all inner (Skolem–Noether)
$(\chi, a)$ or $(L/F, \sigma, a)$	cyclic algebra, $z^n = a$, $z\ell = \sigma(\ell)z$
$H^2(G, L^\times)$	second cohomology, $\cong \operatorname{Br}(L/F)$ for $G = \operatorname{Gal}(L/F)$
$\operatorname{inv}_v(D)$	Hasse invariant at a place $v$