Topics and remarks

General references

• [DuFo04]

• [HoKu71]

• [Hung74]

• [Smit96], whose page links Smith’s course notes.

Group theory

References: [DuFo04]; [Hung74]; [Smit96].

Topics

See [DuFo04].

• Subgroups, left and right cosets, and quotient groups

• The subgroup generated by a subset

• Normalizers

• Lagrange’s theorem

• The isomorphism theorems

• The structure theorem for finite abelian groups

• Direct and semidirect products

  • Recognition of internal direct products

  • Recognition of semidirect products

• Structure of special classes of groups:

  • \(p\)-groups

  • Cyclic groups

  • Dihedral groups

  • Free groups, generators, and relations

  • Symmetric and alternating groups, and cycle decompositions

• Group actions and their applications to the structure of groups:

  • The Sylow theorems and their proofs

  • The orbit-stabilizer theorem

  • The action of \(G\) on the left cosets of a subgroup

  • The action of \(G\) on itself by conjugation

  • The class equation

  • Cayley’s theorem

• There is no nonabelian simple group of order strictly between \(60\) and \(168\)

• \(A_n\) is simple for \(n \geq 5\)

• A subgroup of index \(p\), the smallest prime dividing \(\size G\), is normal

• Groups of order \(p^2\) are abelian

• Automorphisms and inner automorphisms

• Classification of groups of order \(pq\)

• The commutator subgroup

• Solvable groups, nilpotent groups, and composition series

• Upper central series, lower central series, and derived series

• Frattini’s argument

• The Jordan–Hölder theorem

Linear algebra

References: [DuFo04]; [HoKu71]; [Smit96].

Topics

• Determinants

• Eigenvalues and eigenvectors

• The Cayley–Hamilton theorem

• Canonical forms of matrices

  • Jordan canonical form

  • Rational canonical form

• Linear groups (\(\GL_n\), \(\SL_n\), \(\operatorname{O}_n\), \(\operatorname{U}_n\))

• Duality

  • Dual spaces

  • Dual bases

  • The dual of a linear map

  • Double duals

• The spectral theorem in finite dimensions

Rings

References: [DuFo04]; [Hung74]; [Smit96].

• [DuFo04]

• [Smit96]:

  • 843-2, sections 11, 12, and 16-21 (39 pages)

  • 844-1, sections 7-9 (20 pages)

  • 844-2, sections 10-16 (37 pages)

Topics

• Ideals and quotient rings

• The isomorphism theorems for rings

• An ideal \(I\) of a commutative ring \(R\) is maximal if and only if \(R/I\) is a field

• Zorn’s lemma

  • Every vector space has a basis

  • Maximal ideals exist

  • Algebraic closures of fields exist

  • Existence of maximal ideals in countable or Noetherian rings without Zorn’s lemma

  See [Smit96].

• The Chinese remainder theorem

• The Euclidean algorithm

• Prime and irreducible elements

• The Gaussian integers and factorization in \(\ZZ[i]\)

• Localization of a domain and the field of fractions

• Factorization in domains

• Characterizations and properties of classes of rings:

  • Euclidean domain \(\implies\) PID \(\implies\) UFD

  • Integral domains: prime elements are irreducible

  • UFDs: greatest common divisors exist, and a UFD need not be a PID

  • PIDs

    • PIDs are Noetherian

    • Irreducible elements are prime

    • PIDs are UFDs, and greatest common divisors exist

    • Results about PIDs [DuFo04]

      • A PID that is not a Euclidean domain [DuFo04]

      • Proof that a Euclidean domain is a PID and hence a UFD

      • Proof that \(\ZZ\) and \(k[x]\) are UFDs [Smit96, DuFo04]

      • A polynomial ring in infinitely many variables over a UFD is a UFD [DuFo04]

  • Euclidean domains are PIDs

• Gauss’s theorem on unique factorization of polynomials:

  • \(\ZZ[x]\) is a UFD

  • \(R[x]\) is a UFD when \(R\) is a UFD

• Polynomial rings

• Polynomials

  • Gauss’s lemma

  • The remainder and factor theorems

  • Eisenstein’s criterion [DuFo04]

  [DuFo04] states Eisenstein’s criterion for monic polynomials; the proof of the general case is the same, and [Smit96] states the general case.

  • Reducibility

  • The rational root test

• The structure of \(\unitsof{(\ZZ/n\ZZ)}\) as a product of cyclic groups

  See [DuFo04] and [Smit96].

• Gröbner bases and division algorithms for polynomials in several variables [DuFo04]

Modules

References: [DuFo04]; [Hung74]; [Smit96].

Topics

• The isomorphism theorems for modules

• Applications to the structure of:

  • Finitely generated abelian groups

  • Canonical forms of matrices

• Classification of finitely generated modules over PIDs, in particular over Euclidean domains

• Modules over PIDs and canonical forms of matrices [DuFo04 and 12.1-12.3]

  • Constructive proof of the decomposition [DuFo04, exs. 12.1.16-19]

  [Smit96 and 845-2] discusses the constructive proof in detail.

Field theory

References: [DuFo04]; [Hung74]; [Smit96].

Topics

• Algebraic extensions of fields

• Finite fields

• Separable extensions

• The fundamental theorem of Galois theory

• Computation of Galois groups

  • of polynomials of small degree

  • of cyclotomic polynomials

• Solvability of polynomials by radicals