Rings and Fields

General Information | Syllabus | HW Assignments |
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Lectures | Sections in Text | Brief Description |
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Week 1 | 4.4, 4.5, 8.1, 8.2 | Mostly a quick review: Rings (examples, properties, homomorphism theorems), modules (vector spaces as k[x]-modules, group rings), polynomial rings in many variables, division algorithm over commutative rings with 1, polynomials versus polynomial maps |

Week 2 | 8.3, 8.4, CRT in 8.6, | irreducibles and prime elements, prime and maximal ideals, UFDs, PIDs, Noetherian rings, Euclidean domains |

Week 3 | 8.5, 8.8, 9.1, 9.9(cyclotomic polynomials) | Gauss's lemma and corollaries, Irreducibility tests, Hilbert's Basis Theorem, Cyclotomic polynomials, finite and algebraic field extensions |

Week 4 | 9.2, 9.6(part),supplementary material | finite, and algebraic extensions, splitting fields. composites and distinguished classes of extensions |

Week 5 | 9.2, 9.3, 9.4, 9.5 | tests for separability, irreducibility of cylotomic polynomials, finite fields, extending embeddings, existence and uniqueness of splitting fields, algebraic closures and uniqueness, compass and straightedge constructions |

Week 6 | 9.6, 9.7 | embeddings, normality for general algebraic extensions, separability, begin Galois theory |

Week 7 | class notes | Equivalent versions of the FTGT for finite extensions, examples: cyclotomic, biquadratic, x^3 - 2, x^4-2 over Q, normality in Galois extensions, composites and liftings of Galois extensions |

Week 8 | class notes | Finite fields, irreducibles over 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 | class notes | Solvability by radicals, Insolvability of the quintic, The general polynomial of degree n, Intro to Algebraic Number Theory |

Week 10 | class notes | Intro to Algebraic Number Theory |

T. R. Shemanske

Last updated June 17, 2015 12:56:58 EDT