I had my algebra qual yesterday. These are the problems along with my solutions. Hopefully they are correct

Some preliminary comments. First, I know my solution to 1(a) is incorrect. Second, I didn’t know how to prove problem 3. I literally wrote down garbage so I am expecting zero points for that one.

1. Let $\mathbb{Q}$ denote the additive group of rational numbers.

a) Prove that every finitely generated subgroup of $\mathbb{Q}$ is cyclic.

Proof. Let $H$ be a finitely generated subgroup of $\mathbb{Q}$ and let $\{h_1,\ldots,h_m\}$ be the generators of $H$. Pick $x\in H$. We want to show that $x$ can be generated by a single $h$ in $H$. Then $x=n(h_1+\ldots+h_m)$ for some positive integer $n$. But, $h_1+\ldots+h_m=\tilde{h}\in H$. Thus, $x=n\tilde{h}$. $\Box$

b) Prove that $\mathbb{Q}$ is not finitely generated.

Proof. Seeking a contradiction, suppose that $\mathbb{Q}$ is finitely generated. Note that $\mathbb{Q}$ is a subgroup of itself. Thus, $\mathbb{Q}$ is a finitely generated subgroup of $\mathbb{Q}$ and by part (a), this means that $\mathbb{Q}$ must be cyclic. This of course is a contradiction because $\mathbb{Q}$ is not cyclic. $\Box$

2. List, up to isomorphism, all abelian groups of order $3528=2^3\cdot3^2\cdot7^2$.

Solution. The partitions of $3$ are $3$, $2+1$, and $1+1+1$. The abelian groups associated with $2^3$ are:

• $\mathbb{Z}_8$
• $\mathbb{Z}_4 \times \mathbb{Z}_2$
• $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$

The partitions of $2$ are $2$ and $1+1$. The abelian groups associated with $3^2$ are:

• $\mathbb{Z}_9$
• $\mathbb{Z}_3 \times \mathbb{Z}_3$

The abelian groups associated with $7^2$ are:

• $\mathbb{Z}_{49}$
• $\mathbb{Z}_7 \times \mathbb{Z}_7$

By the fundamental theorem of finitely generated abelian groups, all abelian groups of order $3528$ are isomorphic to one of the groups listed below and none of the groups is isomorphic to another.

1. $\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_{49}$
2. $\mathbb{Z}_8 \times \mathbb{Z}_9 \times \mathbb{Z}_7 \times \mathbb{Z}_7$
3. $\mathbb{Z}_8 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_{49}$
4. $\mathbb{Z}_8 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_7 \times \mathbb{Z}_7$
5. $\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_9 \times \mathbb{Z}_{49}$
6. $\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_9 \times \mathbb{Z}_7 \times \mathbb{Z}_7$
7. $\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_{49}$
8. $\mathbb{Z}_4 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_7 \times \mathbb{Z}_7$
9. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_9 \times \mathbb{Z}_{49}$
10. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_9 \times \mathbb{Z}_7 \times \mathbb{Z}_7$
11. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_{49}$
12. $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_3 \times \mathbb{Z}_7 \times \mathbb{Z}_7$ $\Box$

3. Let $V$ be a vector space over a field $F$ and let $T:V\to V$ be a linear operator such that $T^2=T$. Prove that $V=\text{ker}(T)\oplus\text{im}(T)$.

Proof. 😦

4. Let $A$, $B$ be $n\times n$ matrices with entries in a field $F$ and suppose that $AB=BA$. Prove that if $B$ has an eigenspace of dimension one, then $A$ and $B$ share a common eigenvector.

Proof. Since $B$ has an eigenspace of dimension one, let $\lambda$ be an eigenvalue of $B$ and let $v$ be the one eigenvector corresponding to $\lambda$. Then we have that $ABv=BAv$ and $ABv=A\lambda v=\lambda Av$. Thus, $BAv=\lambda Av$ which means that $Av$ is an eigenvector of $B$ with corresponding eigenvalue $\lambda$. But $v$ is the only eigenvector corresponding to $\lambda$ so $Av=v$. This shows that $v$ is an eigenvector of $A$ with corresponding eigenvalue $1$. $\Box$

5. Let $D$ be a principal ideal domain. Prove that every proper nontrivial prime ideal is maximal.

Proof. Let $(p)$ be a proper nontrivial prime ideal of $D$. Let $(a)$ be an ideal of $D$ such that $(p)\subseteq(a)$. Then $p=ab$ for some $b$ in $D$. Thus, $ab\in(p)$. Since $(p)$ is prime, $a\in(p)$ or $b\in(p)$.

Case 1: Suppose that $a\in(p)$. Then $(a)\subseteq(p)$ so $(p)=(a)$.

Case 2: Suppose that $b\in(p)$. Then $b=pu$ for some $u$ in $D$. Thus, $p=apu$ implies that $1=au$ and so $a$ is a unit. Thus, $(a)=D$.

We have shown that $(p)=(a)$ or $(a)=D$ which means that $(p)$ is a maximal ideal of $D$. $\Box$

Each problem is worth 5 points, 15 is a passing score. Best case I think my score will be 2+5+0+5+5 which is 17 and a pass. Realistically, I think my score could be 1+5+0+4+5 which is 15 and a pass. But it could be 1+4+0+4+4 which is a 13 and not a pass. I feel very confidant about problems 2, 4, and 5; those should be five points each. But the qual committee can be very particular about grading. I also feel very confidant about 1(b); I just don’t know how many points that one is worth. All I can do now is wait…