**Topics covered in
the class **
**
Jan. 7: Introduction & algebraic elements.
Jan. 9: transcendental elements and extension fields.
Jan. 14: algebraic elements and extension fields; adjoining roots.
Jan. 16: explicit realization of F(\alpha); multple roots; isomorphisms of field extensions;
Jan. 21: degree of field extensions; algebraic extensions; compass and straightedge construction
Jan. 23: compass and straightedge construction; doubling a cube and
trisecting an angle.
Jan. 28: finite fields.
Jan. 30: finite fields, # of irreducible polynomials.
Feb. 4: transcendental extension, transcendence basis.
Feb. 6: algebraic closed fields.
Feb. 11: primitive elements and primitive element theorem
Feb. 13: extensions of a homomorphism
Feb. 17-20: reading week.
Feb. 24: review of test #1, extension of a homomorphism, finite normal extension, uniqueness of splitting fields.
Feb. 27: Examples of Galios groups, splitting fields of irreducible polynomials,
Mar. 4: Galois group of finite extension, finite normal extension.
Mar. 6: Galois correspondence theorem.
Mar. 11: Galois correspondence theorem.
Mar. 13: cyclotomic extensions.
Mar. 18: cyclotomic polynomials and regular $n$-gon
Mar. 20: Kummer extensions.
Mar. 25: Solvable groups and solvable extensions.
Mar. 27: solvable groups and radical extensions.
April 1: example of unsolvable quintic equation; # of real roots; Cardano's formula.
April 3: Extensions of degree 4
April 8: Extensions of finite fields.
** |