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    Algebra prelims, 2003–2009 (compiled scan)

    SRC-ART-ALG-2003-2009-PRELIMS

    Area
    Algebra
    Status
    draft
    Source
    Contents

    75 problems.

    ALGEBRA QUALIFYING EXAM, SUMMER 2009 (pp. 1–2)

    1. Sylow \(2\)-subgroup of \(S_5\), Sylow \(5\)-subgroup of \(S_{15}\), and Sylow \(p\)-subgroups of \(S_p\)
    2. The class equation, and a nontrivial normal subgroup of a \(p\)-group meets the center
    3. Groups of order \(pq\) with \(p<q\) primes are not simple, abelian if \(p\nmid q-1\), and the possibilities when \(p\mid q-1\)
    4. If \(A\) and \(B\) are normal and \(G/A\), \(G/B\) are abelian then \(G/(A \cap B)\) is abelian
    5. Examples of vanishing tensors over \(\mathbf{Z}\) and \(\mathbf{Q}\), torsion-free nonfree modules, maximal nonprime ideals, non-Noetherian rings, and non-ideal centers
    6. Quotients of Artinian rings, Artinian domains are fields, and primes are maximal
    7. Groups of order \(120\), \(\mathbb{Q}[x]\)-modules of dimension \(6\) annihilated by \((x-3)^2(x+5)\), and rational canonical forms with that minimal polynomial
    8. Factorizations of \(x^5-1\), \(x^6-1\), and related polynomials over \(\mathbf{C}\), \(\mathbf{Q}\), and finite fields
    9. Splitting field of \(x^4-4x^2-3\) over \(\mathbb{Q}\): degree, Galois group, and subfields
    10. True-or-false: constructible roots, finite-field extensions, and Galois composita
    11. \(L = K(\alpha)\) iff the Galois conjugates of \(\alpha\) are distinct
    12. The sum of algebraic elements is algebraic, and the minimal polynomial of \(\sqrt{5} + 2\sqrt{7}\)

    Preliminary exam in algebra, June 11, 2008, 9 a.m.–noon (pp. 3–6)

    1. Sylow’s theorem, and no group of order \(6545\) is simple
    2. Elementary divisors and invariant factors of \(\mathbf{Z}_{15}\times\mathbf{Z}_{20}\times\mathbf{Z}_9\); abelian groups of order \(2700\)
    3. A \(p\)-group acting on a finite set of order not divisible by \(p\) has a fixed point
    4. Conjugacy class size equals the index of the centralizer; restriction to a subgroup of index \(2\)
    5. Rational canonical forms of \(6\times 6\) matrices over \(\mathbb{Q}\) with minimal polynomial \((x+2)^2(x-1)\)
    6. Every Euclidean domain is a principal ideal domain, and \(\mathbb{Z}[i]/(\alpha)\) has representatives of norm less than \(N(\alpha)\) and is finite
    7. Jordan canonical form of a nilpotent matrix
    8. If \(A \oplus A \cong B \oplus B\) for finitely generated modules over a PID then \(A \cong B\)
    9. Galois groups over \(\mathbb{Q}\) of \(x^4-25\) and \(x^3-x+1\)
    10. Definition of a splitting field, and \([E:F] \le n!\) without the fundamental theorem
    11. Galois groups over \(\mathbb{Q}\) of \(x^4+4\) and \(x^3+x^2-2x-1\)
    12. Every finite group is a Galois group over some extension \(E \supseteq F\)

    ALGEBRA QUALIFYING EXAM, SUMMER 2007 (pp. 19–21)

    1. Sylow theorems; groups of order \(105\) have a normal subgroup of order \(35\), and both Sylow \(5\)- and \(7\)-subgroups are normal
    2. \(\operatorname{Aut}(C_n)\cong(\mathbb{Z}/n\mathbb{Z})^\times\)
    3. If \(G/Z(G)\) is cyclic then \(G\) is abelian
    4. Two \(3\)-cycles in \(S_5\) generate \(A_4\) or \(A_5\)
    5. Examples of maximal ideals, prime non-maximal ideals, torsion-free modules, UFDs, and non-Noetherian rings
    6. The characteristic of an integral domain is zero or prime
    7. The ring of integers of \(\mathbb{Q}(\sqrt{2})\) is a Euclidean domain
    8. Abelian groups of order \(90\), and \(\mathbb{R}[x]\)-modules of dimension \(4\) annihilated by \(x^4-1\)
    9. Splitting field of \(x^3-7\) over \(\mathbb{Q}\): Galois group, subfields, and which are Galois
    10. Splitting field of \(x^5-3\) over \(\mathbb{Q}\): cyclotomic subfield, Galois group over \(\mathbb{Q}(\zeta_5)\), and degree
    11. Elements of a Galois extension of degree \(2^n\) are constructible; a root of an irreducible quartic with splitting-field degree \(24\) is not
    12. Degree of a compositum when the factors have coprime degrees, and a counterexample without coprimeness
    13. A finite field has \(p^n\) elements; subfields of a field of size \(3^{12}\); an infinite field of characteristic \(3\)

    Algebra Qualifying Exam of July 12, 2006 (pp. 23–26)

    1. Class equation and nontrivial center of p-group
    2. Sylow subgroups and simplicity
    3. Subgroups of index the smallest prime divisor are normal
    4. Normal subgroups of p-groups of every order up to p^k
    5. Galois extension definition and fundamental theorem of Galois theory
    6. Splitting field and Galois group of \(x^4-2\) over \(\QQ\)
    7. Galois group of finite field and inseparable splitting field
    8. Nonisomorphism of \(\QQ(\sqrt{2})\) and \(\QQ(\sqrt{3})\)
    9. Structure theorem for finitely generated abelian groups
    10. Principal ideal domains
    11. Non-free submodule of a free module
    12. Rational canonical form over Q

    Algebra Preliminary Examination, Summer 2005 (pp. 27–28)

    1. Sylow theorems, groups of order 35, simplicity of order 24
    2. Structure theorem and abelian groups of order \(360\)
    3. Principal ideal domains and the nonprincipal ideal \((2,x)\subset\ZZ[x]\)
    4. Sylow subgroups and a field tower for \((x^3-2)(x^2-3)\)
    5. Roots of \(x^{80}-1\), \(x^{81}-1\), and \(x^{88}-1\) over \(\mathbb F_{81}\)
    6. Non-similar matrices with same rational eigenvalue

    ALGEBRA PRELIMINARY EXAM – 2004 (pp. 29–38)

    1. Groups of size 15 are solvable; no simple groups of size 36
    2. Every group of size 15 is cyclic
    3. The fifty groups of order \(72\)
    4. Normality at the smallest prime index
    5. A Euclidean domain, a UFD that is not a PID, and a domain that is not a UFD
    6. A cyclic \(2\times2\) matrix and canonical forms with characteristic polynomial \((x^2+1)(x+1)^2\)
    7. Local rings characterized by their nonunits
    8. Splitting field of \(x^4-5\) over \(\QQ\)
    9. Every finite group is a Galois group over a suitable fixed field
    10. Non-Galois cubics and cubic subfields of a degree-\(105\) Galois extension
    11. Subfields of a field of order \(5^{20}\) — Fields 4 (PDF p. 38)

    Algebra Preliminary Examination, July 28, 2003 (pp. 39–48)

    1. Sylow subgroups and self-centralizing normalizers at order \(84\)
    2. A normal \(D_8\times D_8\) in every Sylow \(2\)-subgroup of \(S_8\)
    3. No GCD in polynomial ring and subring of Euclidean domain
    4. The prime field and all subfields of \(\mathbb F_{1024}\)
    5. Galois groups of \(x^4+x^3+x^2+x+1\) over \(\QQ\) and \(\mathbb F_5\)
    6. Galois groups with four incomparable proper intermediate fields
    7. Removing a base-field linear factor preserves the splitting field
    8. Abelian groups of order \(720\) and cancellation for finitely generated \(\ZZ\)-modules
    9. Solvability of \(S_4\), quartics, and a quintic counterexample

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