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    Amherst College algebra qualifying-exam study guide (2016)

    SRC-AMHERST-ALGEBRA-STUDY-GUIDE-2016

    Area
    Algebra
    Status
    draft
    Source
    Contents

    31 problems.

    Groups and subgroups

    1. Intersections of subgroups are subgroups — Problem 1
    2. Proper subgroups of a group of order \(pq\) are cyclic — Problem 2
    3. Elements commuting with the cosets of a subgroup form a subgroup — Problem 3
    4. Intersections of normal subgroups are normal — Problem 4
    5. Subgroups containing all commutators are normal — Problem 5
    6. Centralizers are subgroups containing \(\langle s\rangle\) as a normal subgroup — Problem 6
    7. The product \(NH\) of a normal subgroup and a subgroup is a subgroup — Problem 7
    8. \(x^{42}\in N\) for a normal subgroup \(N\) of index 42 — Problem 8
    9. Homomorphisms from a group of order 144 to a group of order 25 are trivial — Problem 9
    10. Cosets of the preimage of a subgroup under a homomorphism — Problem 10
    11. The order of a homomorphic image of a subgroup divides \(|G_1|\) — Problem 11

    Permutations

    1. Recovering a lost permutation in \(A_{10}\) and its order — Problem 12
    2. Disjoint cycles, order and parity of \((4\ 2\ 1)(6\ 1\ 3\ 2)\) — Problem 13
    3. Elements of order 21 and 11 in \(S_{10}\) — Problem 14
    4. Homomorphisms from \(S_n\) to groups of odd order are trivial — Problem 15

    Rings and ideals

    1. Definition of an ideal of a ring — Problem 16
    2. Ideals of \(\mathbb Z\) containing relatively prime integers — Problem 17
    3. An ideal in the ring of upper triangular \(2\times 2\) real matrices — Problem 18
    4. Annihilators are ideals — Problem 19
    5. The ideal \((6,x+1)\) in \(\mathbb Z[x]\) — Problem 20
    6. Quotients by ideals containing all commutators are commutative — Problem 21
    7. Reduced quotient rings and radical ideals — Problem 22
    8. The kernel of a ring homomorphism restricted to an ideal is an ideal — Problem 23
    9. Ideals of a field are trivial — Problem 24
    10. Ideals properly containing an ideal with field quotient contain 1 — Problem 25

    Polynomial rings

    1. Polynomials vanishing at 2 and 5 form an ideal of \(\mathbb R[x]\) — Problem 26
    2. Cosets of a principal ideal in \(F[x]\) and remainder representatives — Problem 27
    3. Cubic polynomials over a field are irreducible exactly when rootless — Problem 28
    4. The field \(\mathbb F_3[x]/(x^2+1)\) and the inverse of \(x\) — Problem 29
    5. Reducibility of \(X^4+X^2+1\) over \(\mathbb F_2\) and irreducibility of \(X^3+2X^2+2X+3\) over \(\mathbb F_5\) — Problem 30
    6. Reducibility of \(x^2+3\) over \(\mathbb F_7\) and an intermediate ideal — Problem 31

    Browse these problems in source order