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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.