Special classes of algebras

$A(\alpha , \beta )$ is an (associative) central simple algebra over the field $F$;

the map

is an involution (i.e. for any $x, y \in A(\alpha , \beta )$ the equalities $\overline{x+y}=\bar{x}+\bar{y}, \overline{x y}=\overline{y x}, \overline{\bar{x}}=x$ hold);

for any $x \in A(\alpha , \beta )$

where $\operatorname {tr} x=x+\bar{x}$ and $N(x)=x \bar{x}$ are elements of the field $F$;

the algebra $A(\alpha , \beta )$ is a division ring if and only if the norm equation $N(x)=0$ has only the zero solution in it;

the algebra $A(\alpha , \beta )$ is either a division ring or is isomorphic to the algebra of matrices $\mathbf{M}_{2}(F)$ — according to whether it has zero divisors or not;

if the norm equation has a nonzero solution in the algebra $A(\alpha , \beta )$, then it also has a solution among the nonzero pure quaternions;

the subalgebra $F(a)$ generated by an element $a$ of the algebra $A(\alpha , \beta )$ is a commutative algebra of dimension $\leqslant 2$ over $F$, and if $a$ is not a zero divisor, then $F(a)$ is a field isomorphic to the splitting field of the polynomial $x^{2}-(\operatorname {tr} a) x+N(a)$;

(Witt's theorem) the norm $N(x)$ is a quadratic form of rank 3 on the space of pure quaternions, and, conversely, to every quadratic form of rank 3 on a three-dimensional vector space $W$ over the field $F$ there corresponds an algebra of generalized quaternions, defined as the vector space $F \oplus W$ with multiplication rules

where $Q$ is the bilinear form on $W$ associated with the given quadratic form, and $\left[w_{1}, w_{2}\right]$ is the vector product of elements of the space $W$;

the construction just given establishes a bijective correspondence between quaternion algebras over the field $F$ (up to isomorphism) and equivalence classes of quadratic forms of rank 3 on a three-dimensional vector space over $F$. (Forms $Q: W \times W \rightarrow F$ and $Q^{\prime }: W^{\prime } \times W^{\prime } \rightarrow F$ are called equivalent if there exist an isomorphism $\alpha : W \rightarrow W^{\prime }$ and an element $\lambda \in F^{*}$ such that $Q^{\prime }(\alpha (x), \alpha (y))=\lambda Q(x, y)$ for all $x, y \in W$.)

the quotient algebra $\mathbb {C}\left[x\right] /\langle f(x)\rangle$ is semisimple if and only if the polynomial $f(x)$ has no multiple roots;

the algebra generated by the field $\mathbb {C}$ and a matrix $A$ in the algebra $\mathbf{M}_{n}(\mathbb {C})$ is semisimple if and only if the minimal polynomial of the matrix $A$ has no multiple roots;

a finite-dimensional algebra over a field is semisimple if and only if it is completely reducible on the left;

a commutative semisimple algebra with unity is isomorphic to a direct sum of fields;

if all idempotents of a semisimple algebra lie in the center, then the algebra is a direct sum of several division rings.

Prove that if $H=0$, then $\mathbb {C}(F, H) \simeq \Lambda (V)$.

The even Clifford algebra $\mathbb {C}^{+}(F, H)$ (or $\mathbb {C}_{Q}^{+}(F)$) is the subalgebra of the Clifford algebra generated by the elements $e_{i_{1}} \ldots e_{i_{2 m}}$ $(m=0,1, \ldots ,[n / 2])$. Prove that the even Clifford algebra of the quadratic form

that does not split into linear factors over $F$, is a quadratic extension of the field $F$, isomorphic to the splitting field

of the form $Q$.

Prove that, for char $F \neq 2$, the even Clifford algebra of a quadratic form $Q$ on a three-dimensional vector space $V$ is isomorphic to the algebra of generalized quaternions of the form $Q^{(2)}$ on the three-dimensional vector space $W=\Lambda^{2} V$ (see problem 65.3).

Under the conditions of problem (c), prove that the quadratic form

on the space of pure quaternions is equivalent to the form $\lambda Q\left(\lambda \in F^{*}\right)$.

Prove that for any homogeneous elements $x \in A_{i}, y \in A_{j}$, $z \in A$ we have

An algebra with a $2$-grading for which the homogeneous elements satisfy these relations is called a Lie superalgebra.

Let $V$ be an $n$-dimensional vector space with basis $\left(e_{1}, \ldots , e_{n}\right)$ over a field $K$ of characteristic other than 2, let $\Lambda (V)$ be the exterior algebra on $V$, let $I$ be the identity operator on $V$, let $L_{0}=K \cdot I$, and let $L_{1}$ be the linear span of the operators $\varphi_{i}$ and $\psi_{i}$, where

Prove that $L=L_{0} \oplus L_{1}$ is a Lie superalgebra with respect to the operation introduced in (a).

for any polynomial $f(x) \in \mathbb {Q}\left[x\right]$ of degree $n$ there is a matrix $A$ of order $n$ such that $f(A)=0$;

the algebra $\mathbf{M}_{n}(\mathbb {Q})$ contains a subalgebra isomorphic to $K$;

if $L$ is a subalgebra of $\mathbf{M}_{n}(\mathbb {Q})$ that is a field, then

$\mathbb {Z}$;

$\mathbb {Z}\left[x\right]$;

$\mathbb {Z}\left[x_{1}, x_{2}, \ldots , x_{n}\right]$.

the identity $x \circ y=-y \circ x$ holds in a Lie ring;

the vectors of three-dimensional space form a Lie ring with respect to addition and the vector product;

every ring $R$ is a Lie ring with respect to addition and the operation $x \circ y=x y-y x$;

the set of all derivations of a ring $R$ is a Lie ring with respect to addition and the operation $D_{1} \circ D_{2}=D_{1} D_{2}-D_{2} D_{1}$.

$\mathbb {Z}_{p}$ is a ring without zero divisors, containing $\mathbb {Z}$;

the element $\sum_{i \geqslant 0} a_{i} p^{i}$ is invertible in $\mathbb {Z}_{p}$ if and only if $a_{0}=1,2, \ldots , p-1$;

the natural homomorphism of groups of invertible elements $\mathbb {Z}_{p}^{*} \rightarrow \mathbf{Z}_{p^{n}}^{*}$ is surjective for every $n$;

every ideal of $\mathbb {Z}_{p}$ is principal and has the form $\left(p^{n}\right), n \geqslant 0$;

find all prime elements of $\mathbb {Z}_{p}$.

Prove that the field of $p$-adic numbers $\mathbb {Q}_{p}$, i.e. the field of fractions of $\mathbb {Z}_{p}$, consists of elements of the form $p^{m} h$, where $m \in \mathbb {Z}, h \in \mathbb {Z}_{p}$.

Show that $\mathbb {Q}$ is contained in $\mathbb {Q}_{p}$ as a subfield.

Prove that an element $p^{m}\left(\sum_{i \geqslant 0} a_{i} p^{i}\right)$ of $\mathbb {Q}_{p}$, where $0 \leqslant a_{i} \leqslant p-1$, lies in $\mathbb {Q}$ if and only if, starting from some $N$, the elements $a_{i}, i \geqslant N$, form a periodic sequence.

Find the images in $\mathbb {Q}_{5}$ of the elements $2 / 7$ and $1 / 3$.