Outline                                                            Definitions
-------                                                            -----------
Algebra (mostly background)
  - rings and fields                                          commutative ring
                                                               integral domain
                                                                         field
                                                                prime subfield
                                                                characteristic
                                                            field of fractions

  - polynomial rings                                                polynomial
    . K[t] has unique factorisation                      polynomial ring, R[t]
    . Euclidean algorithm                    field of rational functions, K(t)
    . Gauss's Lemma
    . Eisenstein's Criterion
    . reduction mod p

Field extensions
  - definition                                                       extension
  - adjoining elements to a field                                         K(A)
                                                              simple extension
  - algebraic elements/extensions             algebraic (element or extension)
                                         transcendental (element or extension)
                                                            minimal polynomial
  - constructing simple extensions with
      a given minimal polynomial: K[t]/(f)
  - isomorphic extensions
    . if alpha and beta are algebraic with the
        same minimal poly. then the extensions
        K(alpha) and K(beta) are isomorphic
    . if alpha is transcendental then K(alpha) is
        iso. to K(t) (the field of rational functions)
  - degree of an extension                                        degree [L:K]
    . Tower Law: if M is intermediate between
      K and L then [L:K] = [L:M][M:K]
  - finite/finitely generated extensions          finitely generated extension
    . finite <==> algebraic & finitely generated              finite extension

Compass and straightedge constructions
  - (x,y) is constructible iff [Q(x,y):Q] is a power of 2
  - famous impossible constructions, e.g.
      duplicating the cube, trisecting angles, squaring the circle

Galois group & Galois correspondence
  - definition and basic properties                       automorphism, Aut(L)
                                                      Galois group, Gamma(L/K)
                                                              Galois extension
                                                                   fixed field
  - splitting fields                                           f splits over L
    . splitting fields for f exist                             splitting field
      and have degree at most (deg f)!                        normal extension
    . splitting fields unique up to                                  separable
      isomorphism
  - normality and separability
    . Galois <==> normal & separable
    . L/K separable ==> L/M and M/K separable
        for any intermediate field M
    . L/K normal ==> L/M normal
        but not necessarily M/K
    . examples of non-normal extensions
        e.g. Q(cube root of 2)/Q
    . L/K normal <==> splitting extension
    . any finite extension is contained in a
        finite normal extension
  - criteria for L/K to be Galois
    . normal and separable (definition)
    . |Gamma(L/K)| = [L:K] (< if not Galois)
    . fixed field of Gamma(L/K) is K (strictly larger if not Galois)

  - Galois correspondence (a.k.a. fundamental theorem)
    . inclusion-reversing maps
        M --> Gamma(L/M) and G --> phi(G)
    . [L:M] = |Gamma(L/M)| and [M:K] = |Gamma(L/K)|/|Gamma(L/M)|
    . M/K normal iff Gamma(L/M) is a normal subgroup of Gamma(L/K)
        and in this case, Gamma(M/K) iso. to Gamma(L/K)/Gamma(L/M)
    . lattices of subgroups and subfields

Galois group of a polynomial
  - definition                                               Galois group of f
  - Galois group of f permutes roots of f                     transitive group
      and may be identified with a subgroup
      of the symmetric group of the roots
  - if f is irreducible then the Galois group
      is transitive on the roots
  - case study: cubic polynomials
    . any irredicuble cubic has Galois group iso. to S3 or A3
    . the discriminant determines which
    . discriminant of t^3+pt+q is -4p^3-27q^2

Solubility
  - definition and basic properties of soluble groups            soluble group
    . G soluble ==> H and G/K soluble for any
       subgroup H and normal subgroup K
    . K and G/K soluble ==> G soluble
    . S_n and A_n are not soluble for n >= 5
  - radical extensions                                       radical extension
    . If L/K Galois then Gamma(L/K) soluble
      <==> L contained in a radical extension
   . f soluble by radicals <==> Galois group of f is soluble
   . examples of non-soluble polynomials

Finite fields
  - exactly one field (up to isomorphism)                        Frobenius map
      of every prime power order q = p^n
  - multiplicative group is cyclic of order q-1
  - Galois group is cycle of order n,
      generated by Frobenius automorphism x --> x^p