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    Hungerford, Algebra

    SRC-TEXT-HUN74

    Area
    Algebra
    Status
    draft
    Source
    Contents

    67 problems.

    I.6

    1. Conjugating a cycle relabels its entries
    2. Two two-generator descriptions of \(S_n\)

    I.9

    1. Free groups are torsion-free
    2. The subgroup generated by \(n\)th powers is normal in a free group — Hungerford 1.9.3

    II.1

    1. Finitely generated torsion-free abelian groups are free
    2. \((\QQ,+)\) is torsion-free but neither finitely generated nor free

    II.2

    1. Noncyclic finite abelian groups contain \(\ZZ_p^2\)
    2. Abelian groups of orders \(64\) and \(96\)

    II.4

    1. Conjugation action of \(G/A\) on a normal abelian subgroup
    2. Cyclic central quotient forces commutativity

    II.5

    1. Normal \(p\)-subgroups lie in every Sylow \(p\)-subgroup
    2. Normal Sylow subgroups split a finite group as their direct product
    3. Unique normal subgroup of index \(q\) when \(|G|=p^n q\) with \(p>q\) — Hungerford 2.5.9

    II.6

    1. Center and central quotient of \(Q_8\)
    2. Groups of orders \(18\), \(20\), and \(30\)

    II.7

    1. Solvability and nonnilpotence of small symmetric groups

    II.8

    1. Extending a quotient composition series by a simple normal subgroup
    2. Groups of order \(p^2q\) are solvable

    IV.1

    1. Ideal multiples and quotient-module scalar action
    2. Simple modules are cyclic, and Schur’s lemma
    3. \(\operatorname{Hom}_R(A,B)\) is an abelian group, \(\operatorname{End}_R(A)\) is a ring, and \(A\) is an \(\operatorname{End}_R(A)\)-module — Hungerford 4.1.7
    4. Injective and surjective forms of the five lemma

    IV.2

    1. Quotient and torsion vector spaces over \(R/(p)\)
    2. \(\dim V^m = m\dim V\)
    3. Rank is additive under direct sums of free modules

    IV.4

    1. \(\operatorname{Hom}(\ZZ_m,A)\cong A[m]\) and duals of \(\ZZ_m\)
    2. \(\operatorname{Hom}(\ZZ_2,-)\) sends the epimorphism \(\ZZ\to\ZZ_2\) to the zero map
    3. \(\operatorname{Hom}_R(R,R)\cong R^{\mathrm{op}}\)
    4. Naturality of the map \(A\to A^{**}\) into the double dual

    IV.6

    1. Free modules over domains are torsion-free, but not conversely
    2. Multiplication and annihilator submodules of a cyclic PID module
    3. Invariant factors of a direct sum of two cyclic PID modules

    V.1

    1. Basic divisibility properties of finite field extensions
    2. Odd-degree elements generate the same field as their squares
    3. Degrees and bases of \(\QQ(\sqrt2,\sqrt3)\) and \(\QQ(i,\sqrt3,\zeta_3)\)
    4. \(\QQ(i)\) and \(\QQ(\sqrt2)\) as vector spaces and fields

    V.3

    1. Degree of a splitting field divides the factorial of the polynomial degree
    2. Algebraic elements inside an algebraically closed field form an algebraic closure
    3. No finite field is algebraically closed
    4. Separability descends through intermediate fields
    5. Finite Galois extensions as separable splitting fields

    V.4

    1. Removing repeated roots without changing the Galois group
    2. Galois groups of \(x^4-5\), \(x^3-2\), and their quadratic composite
    3. Galois correspondence for \((x^3-2)(x^2-3)\)
    4. Irreducible quartics with two real roots have Galois group \(S_4\) or \(D_4\)

    V.5

    1. Fermat’s little theorem \(a^p=a\) in \(\ZZ_p\)
    2. Unique \(p\)th roots in a field of order \(p^n\)
    3. Every finite-field element is a sum of two squares

    V.6

    1. Division by integers prime to the characteristic stays in the base field
    2. Degree prime to the characteristic implies separability
    3. An irreducible polynomial whose degree is prime to the characteristic is separable

    V.8

    1. \(\phi(n)\) is even for \(n>2\), and \(\phi(n)=2\) exactly for \(n=3,4,6\)
    2. Degree of the maximal real cyclotomic subfield

    V.9

    1. A radical extension is radical over every intermediate field
    2. Nonsolvable normal closure of a degree-\(n\) root field

    VII.1

    1. The center of \(M_n(R)\) is \(Z(R)I_n\)
    2. Sums, products, and transposes of symmetric and skew-symmetric matrices
    3. Similarity and matrix equivalence are equivalence relations

    VII.2

    1. One-sided inverses and matrix rank
    2. Rank criteria for consistency and uniqueness of linear systems
    3. Smith normal form as a canonical matrix form

    VII.3

    1. Alternating and skew-symmetric multilinear forms
    2. Only finitely many scalar shifts make \(A+rB\) singular

    VII.4

    1. Degree of the minimal polynomial is bounded by the dimension
    2. Rational and Jordan canonical forms with minimal polynomial \((x-2)^2(x+3)\) or \((x^2+1)(x-7)\)

    VII.5

    1. Cayley-Hamilton theorem for finite free modules
    2. Commuting diagonalizable endomorphisms are simultaneously diagonalizable

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