Rings and Fields

General Information | Syllabus | HW Assignments |
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Lectures | Sections in Text | Brief Description |
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Week 1 | Chapter II | Mostly a quick review: Rings (examples, properties, homomorphism theorems), modules (vector spaces as $k[x]$-modules, group rings), polynomial rings in several variables, division algorithm over commutative rings with 1, polynomials versus polynomial maps, prime and maximal ideals, |

Week 2 | II.2, II.5, IV.1, IV.2 | operations on ideals, correspondence theorem, CRT, irreducibles and prime elements, UFDs, PIDs, Noetherian rings, Euclidean domains |

Week 3 | IV.2, IV.3, IV.4, random tidbits | Gauss's lemma and corollaries, Irreducibility tests, Hilbert's Basis Theorem, Cyclotomic polynomials, start finite and algebraic field extensions |

Week 4 | V.1, V.3 (part),supplementary material | finite, and algebraic extensions, splitting fields, composites and distinguished classes of extensions |

Week 5 | IV.1, V.2, supplementary material | tests for separability, irreducibility of cylotomic polynomials, finite fields, extending embeddings, existence and uniqueness of splitting fields, algebraic closures and uniqueness |

Week 6 | V.2, V.3, V.4, supplementary | compass and straightedge constructions, embeddings, normality for general algebraic extensions, separability, begin Galois theory |

Week 7 | V.4, VI.1 | Equivalent versions of the FTGT for finite extensions, examples: cyclotomic, biquadratic, $x^3-2$, $x^4-2$ over $\mathbb Q$, normality in Galois extensions, composites and liftings of Galois extensions |

Week 8 | VI.1, VI.2,VI.4, VI.6 | Finite fields, irreducibles over $\mathbb F_p$, finite abelian groups are Galois groups, prime cyclotomic fields and primitive elements, Artin's theorem on characters, norm and trace, solvability by radicals, Hilbert's Theorem 90 |

Week 9 | VI.2 - VI.7, class notes | Solvability by radicals, Insolvability of the quintic, The general polynomial of degree $n$ |

T. R. Shemanske

Last updated July 18, 2017 09:28:14 EDT