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 $k$.