#### Syllabus

Week    Sections in Texbooks    Brief Description
Week 1 [S] 7.2-7.3, 7.6
[L] II.1 - II.2 (II.3); IV.1
[J] Chapters 16-17
Overview of the spectrum: integral domains, UFDs, Noetherian domains, PIDs, Euclidean domains. Characteristic of a ring, prime subfield, polynomials in one and several variables and their universal mapping properites, general division algorithm, roots of polynomials, prime and maximal ideals, operations on ideals.
Week 2 [S] 8.1 (8.5)
[L] II.5, IV.1-3
Correspondence theorem, CRT, irreducible and prime elements, UFDs, PIDs, Noetherian rings, Gauss's lemma and corollaries, irreducibility tests?
Week 3 IV.3, IV.4, V.1 Irreducibility tests, cyclotomic polynomials, begin field theory
Week 4 V.1, V.2 Finite and algebraic extensions, splitting fields, composite fields, distinguished classes of extensions
Week 5 IV.1, V.1, V.2 Irreducibility of cyclotomic polynomials, extending embeddings, uniqueness of splitting fields, separability, finite fields, algebraic closure and algebraically closed fields.
Week 6 V.2, V.3, V.4 Construction and uniqueness of algebraic closures (summarized), separability, separable degree, primitive element theorem, being Galois theory.
Week 7 VI.1, supplements Galois correspondence, equivalent versions of correspondence for finite extensions, examples.
Week 8 VI.1-VI.7 More examples, Normality of intermediate extensions, Galois extensions under lifts and composites, counting irreducibles in $\mathbb F_p$, Artin's theorem on characters.
Week 9 VI.2-VI.7 Norm and Trace, Hilbert's Theorem 90, Cyclic and radical extensions, solvability by radicals
3/12 Final Exam 8-11am

Thomas R. Shemanske
Last updated February 25, 2018