189-571B: Higher Algebra II
Assignment 4. Due: Wednesday, March 27.
1.
Let $G$ be a finite group and $F$ a field. Show that the group
algebra $F[G]$ over $F$ is isomorphic to its opposite algebra.
2.
Give an example of an algebra over a field
$k$ which is not isomorphic to its opposite algebra.
3.
Let $A$ be a finite-dimensional
associative $F$-algebra and let $E$ be a field extension of $F$.
Show that the tensor product $E \otimes A$ of $F$-vector spaces has a natural
structure of an $E$-algebra, for which there are natural identifications
$$ {\rm Hom}_{E-{\rm alg}}(E\otimes A, R) = {\rm Hom}_{F-{\rm alg}}(A,R),$$
for all $E$-algebras $R$.
(In categorical language, the functor $A \mapsto E\otimes A$ from the category of
$F$-algebras to the category of $E$-algebras is the adjoint functor
of the "forgetful functor" from $E$-algebras to $F$-algebras
which sends an
$E$-algebra to its underlying $F$-algebra.)
4.
Let $H$ be the ${\mathbb R}$-algebra
of Hamilton quaternions over ${\mathbb R}$.
With notations as in the previous question, show that $\mathbb C \otimes H$
is isomorphic
to the matrix algebra $M_2(\mathbb C)$ over $\mathbb C$.
5.
Let $D_8$ be the dihedral group of order $8$ and let $Q$ be the quaternion
group of order $8$. Show that the group rings $\mathbb C[D_8]$ and
$\mathbb C[Q]$ are isomorphic, but that the group rings with
real coefficients are not isomorphic for these two groups, by writing each of the
associated group rings as a product of central simple algebras over $\mathbb R$.
6.
Let $F$ be a field of characteristic not equal to $2$,
and let $R$ be a non-commutative four-dimensional division
algebra over $F$.
(a)
Show that $R$ contains a
quadratic extension $K$ of $F$, and an element $w$ satisfying $w a = a' w$,
for all $a\in K$, where $a\mapsto a'$ is the non-trivial involution in
${\rm Gal}(K/F)$.
(b) Show that $w^2$ belongs to $F^\times$ and is not the norm of an element
of $K$.
(c) Show that the datum of $K$ and $w^2\in F^\times$
determines the isomorphism
type of $R$ completely.
7.
With notations as in the previous question,
show that the set of isomorphism classes of
non-commutative four-dimensional division
algebras over $F$ containing a given quadratic extension $K/F$ is in bijection
with the non-identity elements of the
group $F^\times/N$ where $N$ denotes the group of norms of non-zero elements of $K$.
Use this to conclude that there are infinitely many pairwise
non-isomorphic division algebras over ${\mathbb Q}$.
8.
Let $A$ be a central simple algebra over $F$ and let $K$ be an $F$-
subalgebra of $A$ that is a field.
(a) Show that the degree $d$ of $K$ over $F$ divides $n$,
where $n^2 := \dim_{F}(A)$.
(b) Let $B$ be the centraliser of $K$ in $A$.
Show that $B$ is a central simple algebra over $K$.
What is its dimension over $K$?
(c) Show that the $K$-algebra $A\otimes_F K$ is isomorphic to
a matrix ring with entries in $B$, and conclude that the class of $A$ and
$B$ are equal in the Brauer group of $K$.
(Hint: observe that $A$ can be equipped with the structure of a
$B^{\rm op}$-module, and construct an explicit $K$-algebra homomorphism
$A\otimes K \rightarrow {\rm End}_{B^{\rm op}}(A)$.)
9.
Problem 10, page 121 of ``Advanced Algebra".
10.
Let $L/k$ be a cyclic cubic extension of $k$, and let
$\sigma$ be a generator of ${\rm Gal}(L/k) = \mathbb Z/3\mathbb Z$.
Fix
an element
$a\in k^\times$
which is not the norm of an element of $L$.
Let $A$ be the $k$-algebra consisting of elements of the form
$$ \lambda_0 + \lambda_1 \theta + \lambda_2 \theta^2, \qquad
\mbox{ where } \lambda_0,\lambda_1, \lambda_{2} \in L,$$
and multiplication is defined by
enforcing the rules
$ \theta \lambda = \sigma(\lambda) \theta,
\mbox{ for all } \lambda\in L, \qquad \qquad \theta^3=a.$
Show that $A$ is a non-commutative division algebra of dimension $9$ over $k$.