**189-571B:** Higher Algebra II

## Assignment 3. Due: Wednesday, February 21.

**1.**
Prove that an integral domain which is an Artinian ring is a field.

**2.**
* This question is embarassingly wrong, so please ignore it in your re-writing of the assignment.*
Let $R$ be an $F$-algebra equipped with a linear functional
$${\rm T}: R \rightarrow F$$
for which the bilinear pairing
$$ \langle \ , \ \rangle: R\times R \rightarrow F, \qquad \langle a,b\rangle := T(ab)$$
is both symmetric and non-degenerate.
Show that $R$ is isomorphic to its opposite algebra $R^{\rm op}$.

**3.** * This old question 3 is not terribly interesting since it is based on a false premise:*
By exhibiting an appropriate function $T$ as in the previous question,
show that the matrix algebra $M_n(F)$ over a field $F$, and the ring
${\mathbb R}[i,j,k]$ of Hamilton quaternions, are both isomorphic to their opposite algebras.
* I propose replacing it with the following question:*
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.

**4.**
Give an example of a ring $R$ which is not isomorphic to its opposite ring.
* We gave an example of such a ring $R$ in class. Try to
produce an example where $R$ is an $F$-algebra.*

**5.** *Those who produced an assignment did
decently on this, but, it would be worth writing up your solution in
a more elegant way by using the universal propert defining the algebra
tensor product.*
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.)

**6.** * This question does not need to be rewritten.*
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 $\mathbb C$.

**7.**
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
rational coefficients are not isomorphic for these two groups.

**8.**
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.

**9.**
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 $K$.

**10.**
Show that there are infinitely many pairwise
non-isomorphic division algebras over ${\mathbb Q}$.