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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 ${\mathbb Q}$.