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189-571B: Higher Algebra II

Assignment 4. Due: Wednesday, March 14.

1. Let $R$ be a central simple algebra over a field $F$, and let $M_1$ and $M_2$ be two $R$-modules that are finite-dimensional over $F$. Show that $M_1$ is isomorphic to an $R$- submodule of $M_2$ if and only if $\dim_F(M_1)\le \dim_F(M_2)$, and that $M_1$ and $M_2$ are isomorphic as $R$-modules if and only if their dimensions over $F$ are equal.

2. Let $E$ be a Galois extension of degree $n$ of a field $F$. Show that the algebra $E\otimes_F E$ is isomorphic to $E^n$. Show that this ceases to be true when $E/F$ is not Galois, by describing the $\mathbb Q$-algebra $\mathbb Q(2^{1/3}) \otimes_\mathbb Q \mathbb Q(2^{1/3})$.

3. Let $A= M_n(F)$ and let $B$ be a commutative $F$-subalgebra of $A$ which is also field. Show by a direct argument (i.e., without using the double centraliser theorem) that the degree $d$ of $B/F$ divides $n$ and that the centraliser of $B$ in $A$ is isomorphic to $C=M_{n/d}(B)$. Check that this verifies the conclusion of the double centraliser theorem, by showing that $B$ is the centraliser of $C$ in $A$ and $\dim_F(A)=\dim_F(B) \times \dim_F(C)$.

4. Let $A= M_n(D)$ where $D$ is a division algebra having a field $F$ as its center, and let $B=D$, viewed as a subalgebra of $A$ by associating to $x\in D$ the corresponding $n\times n$ "scalar matrix". Describe the centraliser $C$ of $B$ in $A$ and compute its dimension over $F$. Use this to verify that $B$ is the centraliser of $C$ in $A$ and that $\dim_F(A)=\dim_F(B) \times \dim_F(C)$, as asserted by the double centraliser theorem.

5. Give an example to illustrate that the hypothesis that $B$ should be simple is crucial for the double centraliser theorem to hold. (I.e., that $B$ need not necessarily be equal to the centraliser of its centraliser in a central simple $F$-algebra $A$, when $B$ is not assumed to be simple.)

6. Let $G_1$ and $G_2$ be finite groups and let $F$ be a field. Show that $F[G_1]\otimes_F F[G_2]$ is isomorphic to the group ring $F[G_1\times G_2]$. Conclude that if $(d_1,\ldots,d_r)$ and $(e_1,\ldots, e_s)$ are a complete list of the dimensions of the distinct irreducible representations of $G_1$ and $G_2$ respectively, then $(d_1e_1, d_1 e_2,\ldots, d_re_s)$ is the full list of the dimensions of irreducible representations of $G_1\times G_2$.

7. Show that all the non-trivial commutative $\mathbb R$-subalgebras of the algebra $\mathbb H$ of Hamilton quaternions are conjugate to each other by an element of $\mathbb H^\times$. Show on the other hand that the algebra $\mathbb Q[i,j,k]$ of Hamilton quaternions over $\mathbb Q$ contains infinitely many pairwise non-isomorphic commutative subalgebras over $\mathbb Q$.

8. Exercise 17 of Higher Algebra, Page 122.

9. Exercise 18 of Higher Algebra, Page 122.

10. Exercise 19 of Higher Algebra, Page 122.