Prof. Michael Thaddeus

Spring 2017

REVIEW SESSION: Saturday, May 6 at 4:10 pm in 602 Hamilton. Bring your questions, which are the only agenda.

FINAL EXAM: Tuesday, May 9 from 4:10-7 pm in 602 Hamilton.

Syllabus

Practice Midterm #1 and
Answers

Midterm Examination #1 and
Answers

Practice Midterm #2 and
Answers

NEW:
Midterm Examination #2,
Answers, and
Letter Grades

NEW:
Practice Final Exam and
Answers

Assignments are due in the collection box marked "Modern Algebra II" outside 417 Mathematics. Make sure you use the right box!

Assignment #1, due Friday,
January 27 at 5 pm.

Assignment #2, due Friday,
February 3 at 5 pm.

Assignment #3, due Friday,
February 10 at 5 pm.

Assignment #4, due Friday,
February 17 at 5 pm.

Assignment #5, due Friday,
February 24 at 5 pm.

Assignment #6, due Friday,
March 10 at 5 pm.

Assignment #7, due Friday,
March 24 at 5 pm.

Assignment #8, due Friday,
March 31 at 5 pm.

Assignment #9, due Friday,
April 7 at 5 pm.

Assignment #10, due Friday,
April 21 at 5 pm.

Assignment #11, due Monday,
May 1 at 5 pm.

Alternative hints for A11#5b: You may or may not wish to use 5a.

Let E be the splitting field of x^{p} - a over F. Show that
the splitting field K of x^{p} -1 is a subfield of E.

Then show that (a) if K = E, then x^{p} - a has a root in F;
(b) if not, then x^{p} - a is irreducible over K, hence over
F.

**Tentative list of topics**

A table is given below of topics to be covered in the course,
together with relevant sections of both recommended
texts. Several disclaimers apply.

(1) The presentation in lecture will not closely follow either
text and may include facts presented in other sections, or not at all, in one text or the other.

(2) The last few topics may or may not be covered, depending on timing.

(3) The lecturer reserves the right to permute or alter the topics at any time!

Topic |
Fraleigh |
Dummit & Foote |
---|---|---|

RINGS | ||

Monoids, rings, and fields | 5.1 | 7.1, 7.2 |

Examples | 5.1 | 7.2 |

Ring homomorphisms | 6.1 | 7.3 |

Subrings and ideals | 6.1 | 7.4 |

Integral domains | 5.2 | 7.1 |

Maximal and prime ideals | 6.2 | 7.4 |

Irreducible and prime elements | 8.3 | |

The division algorithm | 5.6 | 9.2 |

Euclidean and principal ideal domains | 7.2 | 8.2, 8.4 |

F a field implies F[x] a PID | 6.2 | 9.2 |

Gaussian integers | 7.3 | 8.1 |

Euclidean algorithm and GCD | 7.2 | 8.1 |

Irreducible and prime elements in PIDs | 7.1 | 8.3 |

Unique factorization domains | 7.1 | 8.3 |

PID implies UFD | 7.1 | 8.3 |

Existence of GCDs in UFDs | 7.1 | 8.3 |

Gauss's lemma | 7.1 | 9.3 |

R a UFD implies R[x] a UFD | 7.1 | 9.3 |

Eisenstein's criterion | 5.6 | 9.4 |

Prime numbers as sums of squares | 7.3 | 8.3 |

FIELDS | ||

Algebraic elements; minimal polynomial | 8.1 | 13.1, 13.2 |

Multiplicativity of degrees of field extensions | 8.3 | 13.2 |

Ruler and compass constructions | 8.4 | 13.3 |

Existence of splitting fields | 9.3 | 13.4 |

Repeated roots and the derivative | 9.4 | 13.5 |

Finite fields | 8.5 | 13.5 |

Separability, perfect fields | 9.4, 9.5 | 13.5 |

Automorphisms of splitting fields | 9.1 | 14.1 |

The extension theorem | 9.2 | 13.4 |

The Galois group | 9.6 | 14.1 |

Automorphisms of fields and their fixed points | 9.1, 9.6 | 14.1 |

Artin's lemma: [E:F] ≤ # G | ||

Normal and Galois extensions | 9.6 | 13.4, 14.1 |

Equivalent conditions to be Galois | 9.6 | 14.2 |

The fundamental theorem of Galois theory | 9.6 | 14.2 |

Cyclotomic extensions | 9.7, 9.8 | 13.6, 14.5 |

Primitive elements | 9.4 | 14.4 |

Quadratic and other cyclic extensions | 14.5 | |

Kummer's theorem on cyclic extensions | 14.7 | |

Radical and solvable extensions | 9.9 | 14.7 |

Insolvability of the general quintic | 9.9 | 14.7 |