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. ūüė¶

    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…

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