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An Introduction to Homological Algebra

Joseph J. Rotman

Chapter 9

Homology and Groups - all with Video Answers

Educators

Chapter Questions

01:40

Problem 1

Let $E$ be a group of order $m n$, where $(m, n)=1$. Prove that a normal subgroup $K$ of order $m$ has a complement in $E$ if and only if there exists a subgroup $C \subseteq E$ of order $n$. (Kernels in this exercise may not be abelian groups.)

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:02

Problem 2

(Baer). Call a group E injective $^{3}$ in Groups if it solves the obvious universal mapping problem: for every group $G$ and every (not necessarily abelian) subgroup $S \subseteq G$, every homomorphism $f^{\prime}: S \rightarrow E$ can be extended to $G$ :
Prove that $E$ is injective if and only if $E=\{1\}$.
Hint. Let $A$ be free with basis $\{x, y\}$, and let $B$ be the semidirect product $B=A \rtimes\langle z\rangle$, where $z$ is an element of order 2 that acts on A by $z x z=y$ and $z y z=x$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:05

Problem 3

(i) Let $K$ be a $Q$-module, where $Q$ is a group. Prove that $K$ is also a right $\mathbb{Z} Q$-module if one defines $a x$ to be $x^{-1} a$, where $x \in Q$ and $a \in K$.
(ii) If a $Q$-module $K$ is made into a right $\mathbb{Z} Q$-module, as in part (i), give an example showing that $K$ is not a $(\mathbb{Z} Q, \mathbb{Z} Q)$ bimodule.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 4

Give an example of a split extension $0 \rightarrow K \rightarrow G \stackrel{p}{\rightarrow} Q \rightarrow 1$ in Groups for which there does not exist a homomorphism $q: G \rightarrow K$ with $q i=1_{K}$. Compare with Exercise $2.8$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 5

Let $0 \rightarrow B \rightarrow A \rightarrow \mathbb{I}_{p} \rightarrow 0$ be an exact sequence of finite abelian $p$-groups, where $p$ is prime. If $B$ is cyclic, prove that either $A$ is cyclic or the sequence splits.

Wendi Zhao
Wendi Zhao
Numerade Educator
03:13

Problem 6

If $G=K \rtimes Q$ and $Q \subseteq N \subseteq G$, prove that $N=(N \cap K) \rtimes Q$.
Hint. Adapt the proof of Corollary $2.24$.
V}$.

Gideon Idumah
Gideon Idumah
Numerade Educator
05:10

Problem 7

Prove that $Q$, the group of quaternions, is not a semidirect product. Hint. The quaternion group $\mathbf{Q}$ is the subgroup of order 8 ,
$$
\begin{aligned}
\mathbf{Q} &=\left\{I, A, A^{2}, A^{3}, B, B A, B A^{2}, B A^{3}\right\} \\
&=\langle A\rangle \cup B\langle A\rangle \subseteq \mathrm{GL}(2, \mathbb{C})
\end{aligned}
$$
where $I=\left[\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right], A=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$, and $B=\left[\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right]$. Note that $A^{2}=-I$ is the unique element of order 2 and that $Z(\mathbf{Q})=\langle-I\rangle$.

Ely Crowder
Ely Crowder
Numerade Educator
08:25

Problem 8

If $K$ and $Q$ are solvable groups, prove that a semidirect product of $K$ by $Q$ is also solvable.
${ }^{3}$ The term injective had not yet been coined when $R$. Baer, who introduced the notion of injective module, proved this result. After recognizing that injective groups are duals of free groups, he jokingly called such groups fascist groups, and he was delighted to have proved that they are trivial.

Ely Crowder
Ely Crowder
Numerade Educator
01:40

Problem 9

Let $K$ be an abelian group, let $Q$ be a group, and let $\theta: Q \rightarrow$ $\operatorname{Aut}(K)$ be a homomorphism. Prove that $K \rtimes Q \cong K \times Q$ if and only if $\theta$ is the trivial map; that is, $\theta_{x}=1_{K}$ for all $x \in Q$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
06:02

Problem 10

(i) If $K$ is cyclic of prime order $p$, prove that $\operatorname{Aut}(K)$ is cyclic of order $p-1$.
(ii) Let $G$ be a group of order $p q$, where $p>q$ are primes. If $q \nmid(p-1)$, prove that $G$ is cyclic. Conclude, for example, that every group of order 15 is cyclic.

Ely Crowder
Ely Crowder
Numerade Educator
01:57

Problem 11

(i) Prove that $\operatorname{Aut}\left(S_{3}\right) \cong \mathrm{GL}(2,2) \cong S_{3}$.
(ii) Prove that if $G$ is a group, then Aut $(G)=\{1\}$ if and only if $|G| \leq 2$. Conclude that every abelian group of order $>2$ has an outer automorphism.
(iii) Prove that $D_{8}$ has an outer automorphism.
Hint. $D_{8}=\langle a, b\rangle$, where $a^{4}=1=b^{2}$ and $b a b=a^{-1}$.
Define $\varphi: D_{8} \rightarrow D_{8}$ by $\varphi(a)=a^{3}$ and $\varphi(b)=b$.
(iv) Prove that $\mathbf{Q}$ has an outer automorphism.
Hint. Show that $\operatorname{Aut}(\mathbf{Q}) \cong S_{4}$ and $\operatorname{Inn}(\mathbf{Q}) \cong \mathbf{

Wendi Zhao
Wendi Zhao
Numerade Educator
01:02

Problem 12

Let $U:$ Rings $\rightarrow$ Groups be the functor assigning to each ring $R$ its group of (two-sided) units $U(R)$; let $F:$ Groups $\rightarrow$ Rings be the functor assigning to each group $G$ its integral group ring $\mathbb{Z} G$ and to each group homomorphism $\varphi: G \rightarrow H$ the ring homomorphism $F(\varphi): \mathbb{Z} G \rightarrow \mathbb{Z} H$, defined by $\sum_{x \in G} m_{x} x \mapsto \sum_{x \in G} m_{x} \varphi(x)$. Prove that $(F, U)$ is an adjoint pair of functors.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:52

Problem 13

Let $Q$ be a group and let $K$ be a $Q$-module. Prove that any two split extensions of $K$ by $Q$ realizing the operators are equivalent.

Wendi Zhao
Wendi Zhao
Numerade Educator
01:07

Problem 14

Let $Q$ be abelian, let $K$ be a $Q$-module, and let $A(Q, K)$ be the subset of $H^{2}(Q, K)$ consisting of all $[0 \rightarrow K \rightarrow E \rightarrow Q \rightarrow 1]$ with $E$ abelian.
(i) Prove that $A(Q, K)$ is a subgroup of $H^{2}(Q, K)$.
(ii) Prove that $A(Q, K) \cong \mathrm{Ext}_{2}^{1}(Q, K)$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:10

Problem 15

The generalized quaternion group $Q_{n}$, for $n \geq 3$, is the subgroup of $\mathrm{GL}(2, \mathbb{C})$ generated by $A=\left[\begin{array}{cc}0 & \omega \\ \omega & 0\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & 1 \\ -1 & 0\end{array}\right]$, where $\omega$ is a primitive $2^{n-1}$ th root of unity. Note that $\left|\mathbf{Q}_{n}\right|=2^{n}$ and that
$$
A^{2^{n-1}}=1, \quad B A B^{-1}=a^{-1}, \text { and } B^{2}=A^{2^{n-2}}
$$
(i) Prove that $B$ is the unique element of order $2, Z\left(\mathbf{Q}_{n}\right)=$ $\langle B\rangle$, and that $\mathbf{Q}_{n}$ is not a semidirect product.
(ii) Prove that $\mathbf{Q}_{n}$ is a central extension (i.e., $\theta$ is trivial) of $\mathbb{I}_{2}$ by $D_{2^{n-1}}$.
(iii) Using factor sets, give a proof of the existence of $Q_{n}$.

Ely Crowder
Ely Crowder
Numerade Educator
View

Problem 16

If $p$ is an odd prime, prove that every group $G$ of order $2 p$ is a semidirect product of $\mathbb{I}_{p}$ by $\mathbb{I}_{2}$, and conclude that either $G$ is cyclic or $G \cong D_{2 p}$.

Nick Johnson
Nick Johnson
Numerade Educator
01:58

Problem 17

(i) Let $T$ be the subgroup of $\mathrm{GL}(2, \mathbb{C})$ generated by $\left[\begin{array}{cc}\omega & 0 \\ 0 & \omega^{2}\end{array}\right]$ and $\left[\begin{array}{ll}0 & i \\ 1 & 0\end{array}\right]$, where $\omega=e^{2 \pi i / 3}$ is a primitive cube root of unity. Prove that $|T|=12$.
(ii) Prove that $T$ has a presentation
$$
\left(a, b \mid a^{6}=1, b^{2}=a^{3}=(a b)^{2}\right)
$$
(iii) Prove that $T \cong \mathbb{I}_{3} \times \mathbb{I}_{4}$.
Hint. Let $K=\langle u\rangle \cong \mathbb{I}_{3}$, let $Q=\langle x\rangle \cong \mathbb{I}_{4}$, and make $K$ into a $Q$-module by $x u=2 u, x(2 u)=u$, and $x^{2} u=u .$ In $K \times Q$, define $a=\left(2 u, x^{2}\right)$ and $b=(0, x) .$
(iv) Prove that every group $G$ of order 12 is isomorphic to exactly one of the following five groups:
$$
\mathbb{I}_{12}, \quad \mathbf{V} \times \mathbb{I}_{3}, \quad A_{4}, \quad S_{3} \times \mathbb{I}_{2}, \quad T .
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
05:10

Problem 18


(i) Let $A$ and $B$ be $G$-modules. Prove that $\operatorname{Hom}_{\mathrm{Z}}(A, B)$ is a $G$-module under diagonal action:
$$
(g \varphi)(a)=g \varphi\left(g^{-1} a\right)
$$
for all $\varphi: A \rightarrow B, g \in G$, and $a \in A$. Moreover, prove that $\operatorname{Hom}_{G}(A, B)=\operatorname{Hom}_{\mathbb{Z}}(A, B)^{G}$.
(ii) Let $A$ be a right $\mathbb{Z} G$-module, and let $B$ be a (left) $G$-module. Prove that $A \otimes \mathbb{Z} B$ is a $G$-module under diagonal action:
$$
g(a \otimes b)=g a \otimes g a
$$
for all $g \in G, a \in A$, and $b \in B$.

Doruk Isik
Doruk Isik
Numerade Educator
01:40

Problem 19

Let $G$ and $Q$ be groups, and let $\varphi: \mathbb{Z} G \rightarrow \mathbb{Z} Q$ be a ring homomorphism.
(i) Prove that if $K$ is a $Q$-module, then $\varphi$ equips $K$ with the structure of a $G$-module (which we denote by $\varphi K$ ).
Hint. See Proposition 2.1: if $\sigma: \mathbb{Z} Q \rightarrow \operatorname{End}(K)$, then $\varphi \sigma: \mathbb{Z} G \rightarrow \operatorname{End}(K)$.
(ii) If $G$ and $Q$ are groups with isomorphic group rings, $\mathbb{Z} G \cong$ $\mathbb{Z} Q$, prove that $G$ and $Q$ have the same homology and the same cohomology: for every $G$-module $K, H_{n}(G, \varphi K) \cong$ $H_{n}(Q, K)$ and $H^{n}\left(G,{ }^{\varphi} K\right) \cong H^{n}(Q, K)$.
(iii) If $G$ and $Q$ are abelian groups with isomorphic group rings, prove that $G \cong Q$.
Hint. $H_{1}(G, \mathbb{Z}) \cong H_{1}(Q, \mathbb{Z})$.
There exist nonisomorphic finite groups whose integral group rings are isomorphic [see M. Hertweck, "A counterexample to the isomorphism problem for integral group rings," Annals Math 154 (2001), 115-138].

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:53

Problem 20

Let $G$ be a group and, for $n \geq 1$, let $B_{n}$ be the free $G$-module with basis $X=\left\{\left[x_{0}|\cdots| x_{n}\right]: x_{i} \in G\right\}\left(B_{n}\right.$ is the $n$th term of the bar resolution). Prove that $\left(B_{n}\right)_{G}=B_{n} / \mathcal{G} B_{n}$ is the free $\mathbb{Z}$-module with basis $X$.

Victor Salazar
Victor Salazar
Numerade Educator
00:59

Problem 21

Give an example of a group $G$ and a $G$-module $A$ for which Proposition $9.53$ is false; that is, $H_{1}(G, A) \not H_{1}(G, \mathbb{Z}) \otimes_{\mathbb{Z}} A$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:25

Problem 22

Let $G$ be a group, let $k$ be a commutative ring, and let $k G$ be the group algebra.
(i) If $A$ is a left $k G$-module, prove, for all $n \geq 0$, that
$$
\operatorname{Ext}_{k G}^{n}(k, A) \cong \operatorname{Ext}_{Z G}^{n}(\mathbb{Z}, A) .
$$
Conclude that the cohomology groups $H^{n}(G, A)$ do not depend on the coefficient ring $k$.
(ii) If $A$ is a left $k G$-module, prove, for all $n \geq 0$, that
$$
\operatorname{Tor}_{n}^{k G}(k, A) \cong \operatorname{Tor}_{n}^{Z G}(\mathbb{Z}, A) .
$$
Conclude that the homology groups $H_{n}(G, A)$ do not depend on the coefficient ring $k$.

Ely Crowder
Ely Crowder
Numerade Educator
04:42

Problem 23

(i) Prove that $\varphi: \mathbb{Z} G \rightarrow(\mathbb{Z} G)^{\text {op }}$, defined by $\varphi: \sum_{x} m_{x} x \mapsto$ $\sum_{x} m_{x} x^{-1}$, is a ring isomorphism.
(ii) Let $A$ be a left $\mathbb{Z} G$-module and $B$ be a right $\mathbb{Z} G$-module. Prove that $\operatorname{Tor}_{n}^{Z G}(B, A) \cong \operatorname{Tor}_{n}^{Z G}(A, B)$, where $A, B$ are viewed [as in part (i)] as right and left $\mathbb{Z} G$-modules, respectively, in the second Tor. Compare with Theorem 7.1.

Mengchun Cai
Mengchun Cai
Numerade Educator
01:40

Problem 24

For a group $G$ and integer $m>0$, view $\mathbb{I}_{m}$ as a trivial $G$-module. Prove that $H_{1}\left(G, \mathbb{I}_{m}\right) \cong G / G^{\prime} G^{m}$, where $G^{\prime}$ is the commutator subgroup and $G^{m}$ is the subgroup generated by all $m$ th powers.
Hint. Consider the exact sequence of $G$-trivial modules
$$
0 \rightarrow \mathbb{Z} \stackrel{m}{\longrightarrow} \mathbb{Z} \rightarrow \mathbb{I}_{m} \rightarrow 0 .
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
07:44

Problem 25

Let $F$ be a free group. If $I$ and $J$ are two-sided ideals in $\mathbb{Z} F$, which are free $F$-modules on $U$ and $V$, respectively, prove that $I J$ is a free $F$-module with basis $U V=\{u v: u \in U, v \in V\}$.

Anthony Ramos
Anthony Ramos
Numerade Educator
00:59

Problem 26

If $G$ is a group, prove that $P_{n} \cong \bigotimes^{n+1} \mathbb{Z} G$, where $P_{n}$ is the $n$th term in the homogeneous resolution $\mathbf{P}(G)$ and
$\bigotimes \mathbb{Z} G=\mathbb{Z} G \otimes_{\mathbb{Z}} \mathbb{Z} G \otimes_{\mathbb{Z}} \cdots \otimes_{\mathbb{Z}} \mathbb{Z} G$
the tensor product of $\mathbb{Z} G$ with itself $n$ times.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:57

Problem 27

If $G$ is a finite cyclic group, prove, for all $G$-modules $A$ and for all $n \geq 1$, that $H^{n}(G, A) \cong H_{n+1}(G, A)$.

Wendi Zhao
Wendi Zhao
Numerade Educator
00:59

Problem 28

Let $G$ be a finite cyclic group, and let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence of $G$-modules.
(i) Prove that there is an exact hexagon:
(ii) Prove that if the Herbrand quotient is defined for two of the modules $A, B, C$ [that is, both $H^{1}(G, M)$ and $H^{2}(G, M)$ are finite, where $M=A, B$, or $C$ ], then it is defined for the third one, and
$$
h(B)=h(A) h(C)
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:04

Problem 29

If $R=\mathbb{Z}[x] /\left(x^{k}-1\right)$, prove that $\mathrm{D}(R)=\infty$ (where $\mathrm{D}$ is global dimension).

Doruk Isik
Doruk Isik
Numerade Educator
00:59

Problem 30

(Barr-Rinehart) For a group $G$, define $\widetilde{H}^{n}(G, A)=\operatorname{Ext}_{Z G}^{n}(\mathcal{G}, A)$, where $A$ is a $G$-module and $\mathcal{G}$ is the augmentation ideal. Prove that $\widetilde{H}^{0}(G, A) \cong \operatorname{Der}(G, A)$ and $\widetilde{H}^{n}(G, A) \cong H^{n+1}(G, A)$ for $n \geq 1$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:40

Problem 31

Consider the commutative diagram of modules
in which $d \Delta=0, f$ is surjective, and $g$ is injective.
(i) Prove that $\bar{d}: B / \operatorname{im} \Delta \rightarrow C$, given by $b+\operatorname{im} \Delta \mapsto d b$, is a well-defined map with ker $\bar{d}=\operatorname{ker} d / \operatorname{im} \Delta$.
(ii) Prove that $\varphi: B / \operatorname{im} \Delta \rightarrow C^{\prime}$, given by $b+\operatorname{im} \Delta \mapsto \alpha f b$, is well-defined.
(iii) Using surjectivity of $f$, prove that $\operatorname{ker} \bar{d} \cong \operatorname{ker} \alpha$.
(iv) As in the first three parts, prove that coker $\bar{d} \cong$ coker $\alpha$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
01:58

Problem 32

If $T \subseteq S \subseteq G$ are subgroups of finite index, use Theorem $9.97$ to prove
$$
V_{G \rightarrow T}=V_{S \rightarrow T} V_{G \rightarrow S}
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:16

Problem 33

(i) Give an example of a induced $G$-module that is not injective.
(ii) Give an example of an induced $G$-module that is not projective.

Vysakh M
Vysakh M
Numerade Educator
03:38

Problem 34

The ring $\mathcal{L}_{\mathbb{Z}}[x]$ of all Laurent polynomials over $\mathbb{Z}$ in one indeterminate consists of all formal sums $\sum_{i=k}^{n} m_{i} x^{i}$, where $m_{i} \in \mathbb{Z}$ and $k \leq n$ are (possibly negative) integers.
(i) Prove that $\mathcal{L}_{\mathrm{Z}}[x] \cong \mathbb{Z} G$, where $G \cong \mathbb{Z}$.
(ii) If $S=\left\{x^{k}: k \geq 0\right\}$, prove that $\mathcal{L}_{Z}[x] \cong S^{-1} \mathbb{Z}[x]$.
(iii) If $G$ is the free abelian group with basis $\left\{x_{1}, \ldots, x_{n}\right\}$, define the ring $\mathcal{L}_{\mathrm{Z}}\left[x_{1}, \ldots, x_{n}\right]$ of Laurent polynomials over $\mathbb{Z}$ in $n$ indeterminates to be $S^{-1} \mathbb{Z}\left[x_{1}, \ldots, x_{n}\right]$. Prove that $\mathcal{L}_{\mathbb{Z}}\left[x_{1}, \ldots, x_{n}\right] \cong \mathbb{Z} G .$
(iv) Prove that cd $(\mathbb{Z} G) \leq n+1$ when $G$ is free abelian of rank $n$.
Hint. Use Hilbert's Syzygy Theorem.

Wendi Zhao
Wendi Zhao
Numerade Educator
View

Problem 35

Let $G$ be a group. If $B$ and $A$ are $G$-modules, make $A \otimes_{\mathbb{Z}} B$ into a $G$-module with diagonal action:
$$
g(b \otimes a)=(g b) \otimes(g a) .
$$
If $A$ is a $G$-module, let $A_{0}$ be its underlying abelian group. Prove that $\mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \cong \mathbb{Z} G \otimes_{\mathbb{Z}} A$ as $G$-modules.
Hint. Define $f: \mathbb{Z} G \otimes_{\mathbb{Z}} A_{0} \rightarrow \mathbb{Z} G \otimes_{\mathbb{Z}} A$ by $g \otimes a \mapsto g \otimes g a$.

Nick Johnson
Nick Johnson
Numerade Educator
00:59

Problem 36

Let $G$ be a finite group and let $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow 0$ be an exact sequence of $G$-modules. Find a formula for the boundary maps, and prove exactness of
$$
\tilde{H}^{-2}(G, C) \rightarrow \widetilde{H}^{-1}(G, A) \rightarrow \widetilde{H}^{-1}(G, B)
$$
and
$$
\tilde{H}^{0}(G, B) \rightarrow \widetilde{H}^{0}(G, C) \rightarrow \widetilde{H}^{1}(G, A)
$$

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
08:50

Problem 37

Prove that $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H} \cong$ Mat $_{4}(\mathbb{R})$ as $\mathbb{R}$-algebras.
Hint. Every simple left artinian ring, e.g., $\mathbb{H} \otimes_{\mathbb{R}} \mathbb{H}$, is isomorphic to $\operatorname{Mat}_{n}(\Delta)$ for some $n \geq 1$ and some division ring $\Delta$.

Ely Crowder
Ely Crowder
Numerade Educator
08:50

Problem 38

We have given one isomorphism $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H} \cong$ Mat $_{2}(\mathbb{C})$ in Example $9.121$. Describe all possible isomorphisms between these two algebras.
Hint. Use the Skolem-Noether Theorem.

Ely Crowder
Ely Crowder
Numerade Educator
20:16

Problem 39

Prove that $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{C} \cong \mathbb{C} \times \mathbb{C}$ as $\mathbb{R}$-algebras.

Cullen Miller
Cullen Miller
Numerade Educator
08:50

Problem 40

(i) Let $\mathbb{C}(x)$ and $\mathbb{C}(y)$ be function fields. Prove that $R=$ $\mathbb{C}(x) \otimes_{\mathbb{C}} \mathbb{C}(y)$ is isomorphic to a subring of $\mathbb{C}(x, y)$. Conclude that $R$ has no zero-divisors.
(ii) Prove that $\mathbb{C}(x) \otimes_{\mathbb{C}} \mathbb{C}(y)$ is not a field.
Hint. Show that $R$ is isomorphic to the subring of $\mathbb{C}(x, y)$ consisting of polynomials of the form $f(x, y) / g(x) h(y)$.

Ely Crowder
Ely Crowder
Numerade Educator
02:52

Problem 41

Let $A$ be a central simple $k$-algebra. If $A$ is split by a field $E$, prove that $A$ is split by any field extension $E^{\prime}$ of $E$.

Vishnu P
Vishnu P
Numerade Educator
02:36

Problem 42

Let $E / k$ be a field extension. If $A$ and $B$ are central simple $k$ algebras with $A \sim B$, prove that $E \otimes_{k} A \sim E \otimes_{k} B$ as central simple $E$-algebras.

James Chok
James Chok
Numerade Educator
View

Problem 43

Prove that Mat $_{2}(\mathbb{H}) \cong \mathbb{H} \otimes_{\mathbb{R}}$ Mat $_{2}(\mathbb{R})$ as $\mathbb{R}$-algebras.

Victor Salazar
Victor Salazar
Numerade Educator
08:53

Problem 44

{ (i) Let } A \text { be a four-dimensional vector space over } \mathbb{Q} \text {, and let } \\ & 1, i, j, k \text { be a basis. Prove that } A \text { is a division algebra over }\end{array}$ $\mathbb{Q}$ if we define 1 to be the identity and
$$
\begin{array}{lll}
i^{2}=-1, & j^{2}=-2, & k^{2}=-2, \\
i j=k, & j k=2 i, & k i=j, \\
j i=-k, & k j=-2 i, & i k=-j .
\end{array}
$$
(ii) Prove that $\mathbb{Q}(i)$ and $\mathbb{Q}(j)$ are nonisomorphic maximal subfields of $A$.

Victor Salazar
Victor Salazar
Numerade Educator
04:31

Problem 45


Let $D$ be the $\mathbb{Q}$-subalgebra of $\mathbb{H}$ having basis $1, i, j, k$.
(i) Prove that $D$ is a division algebra over $\mathbb{Q}$.
Hint. Compute the center $Z(D)$.
(ii) For any pair of nonzero rationals $p$ and $q$, prove that $D$ has a maximal subfield isomorphic to $Q\left(\sqrt{-p^{2}-q^{2}}\right)$.
Hint. Compute $(p i+q j)^{2}$.

ET
Ed Tam
Numerade Educator
00:27

Problem 46

(Dickson) If $D$ is a division algebra over a field $k$, then each $d \in D$ is algebraic over $k$. Prove that $d, d^{\prime} \in D$ are conjugate in $D$ if and only if $\operatorname{irr}(d, k)=\operatorname{irr}\left(d^{\prime}, k\right)[\operatorname{irr}(d, k)$ is the polynomial in $k[x]$ of least degree having $d$ as a root].
Hint. Use the Skolem-Noether theorem.

Heather Zimmers
Heather Zimmers
Numerade Educator
02:16

Problem 47

Prove that if $A$ is a central simple $k$-algebra with $A \sim$ Mat $_{n}(k)$, then $A \cong \operatorname{Mat}_{m}(k)$ for some integer $m .$

Anthony Ramos
Anthony Ramos
Numerade Educator
02:01

Problem 48

Show that the structure constants in the crossed product $(E, G, f)$ are
$$
g_{\alpha}^{\sigma, \tau}=\left\{\begin{array}{cl}
f(\sigma, \tau) & \text { if } \alpha=\sigma \tau \\
0 & \text { otherwise. }
\end{array}\right.
$$

Christopher Callahan
Christopher Callahan
Numerade Educator