Search the corpus

    Emory qualifying-exam problems by Santiago Arango-Piñeros

    SRC-EMORY-QUAL-PROBLEMS

    Area
    Algebra, Complex Analysis, Real Analysis
    Status
    draft
    Source
    Contents

    37 problems.

    1.1 Groups

    1. Classification of groups of order \(182\)
    2. The normalizer of a subgroup of order \(p^k\) with \(k<n\) in a group of order \(p^nm\) properly contains the subgroup
    3. Index-\(n\) subgroups of \(S_n\) and point stabilizers
    4. Groups of order \(351=3^3\cdot 13\) are not simple; groups of order \(33\) are cyclic
    5. If \(G/Z(G)\) is cyclic then \(G\) is abelian; \(p\)-groups have nontrivial center; groups of order \(p^2\) are abelian
    6. No finite group is the union of conjugates of a proper subgroup; a transitive action on more than one point has a fixed-point-free element
    7. Classification of groups of orders \(15\) and \(30\)
    8. Number of Sylow \(p\)-subgroups of \(S_p\)
    9. A group with at most one subgroup of each order is cyclic, and finite subgroups of \(F^\times\) are cyclic

    1.2 Fields and Galois Theory

    1. Finite-field embeddings and divisibility of extension degrees
    2. A finite field extension of a finite field is Galois with cyclic Galois group
    3. Factorial splitting-field degree forces irreducibility and minimal root fields
    4. Galois groups of \(\mathbb{Q}(\sqrt{2+\sqrt{2}})/\mathbb{Q}\) and of \(x^3-5\)

    1.3 Rings

    1. \(F[[x]]\) is a Euclidean domain, and Euclidean domains are PIDs
    2. The subring of \(F[X]\) of polynomials with vanishing \(X\)-coefficient is not a UFD
    3. \(R/I\) is a field for maximal \(I\); nonzero primes in a PID are maximal; \(R/(p)\) is a field for prime \(p\)

    1.4 Linear Algebra

    1. A square matrix is conjugate to its transpose
    2. Conjugacy classes of \(16\times16\) rational matrices with minimal polynomial \((x^2+1)^2(x^3+2)^2\)
    3. The evaluation map \(V\to(V^\vee)^\vee\) is injective, and an isomorphism if and only if \(V\) is finite-dimensional
    4. The field of fractions of a PID that is not a field is not a finitely generated module
    5. Every vector space has a basis, by Zorn’s lemma

    2.1 Complex Analysis

    1. \(\int_0^\infty \frac{\cos x}{(1+x^2)^2}\,dx\) by residues
    2. Cauchy integral formula for holomorphic functions
    3. The maximum modulus principle from the Cauchy integral formula
    4. An entire function with \(|f(z)|\leq A|z|^2\) is a polynomial of degree at most \(2\)
    5. Schwarz lemma and the Schwarz–Pick inequality
    6. Riemann mapping theorem and uniqueness
    7. A contour through a double pole and a Fourier-type integral
    8. Weierstrass’s theorem on locally uniform limits of holomorphic functions
    9. Nonconstant entire functions have dense image
    10. Rouché’s theorem and local bijectivity where the derivative is nonzero
    11. A meromorphic function on \(\CC\) with \(|f(z)|\to\infty\) as \(|z|\to\infty\) is rational

    2.2 Real Analysis

    1. Carathéodory extension of a measure from an algebra
    2. Fatou’s lemma
    3. Dominated convergence under convergence in measure
    4. \(\lim_{p\to 0^+}\|f\|_p\) exists for \(f\in L^1([0,1])\) and is zero if \(m\{f=0\}>0\)
    5. Absolute continuity and image-measure conditions for a continuous function

    Browse these problems in source order