Math 120b
 
Abstract Algebra
Winter 2014-15
 
MWF 9:00 AM // 257 Sloan
Course Description | Policies | Textbooks | Lecture Notes | Handouts | Homework | Sections

Instructor:  Tom Graber, 362 Sloan, 626-395-4359, graber@caltech.edu
Office Hours:
Monday 10-11
TA: Serin Hong, 382 Sloan, 626-395-4366
Office Hours: Sundays, 3 pm - 4 pm


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Course Description
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The second term of Math 120 will be devoted primarily to Field and Galois Theory. Time permitting, topics should include:

Field extensions:
 Algebraic closure. Splitting fields and normal extensions. Separable and Inseparable extensions and degree. Primitive element theorem. Linear independence of characters. Traces and norms. Normal basis theorem.

Galois theory: Galois extensions. Absolute Galois group. Galois theory for infinite extensions.

Examples: Finite fields. Cyclotomic fields.

Galois Cohomology: Galois Cohomology. Hilbert's 90 theorem.

Kummer Theory: Cyclic extensions. Solvable extensions. Kummer Theory.

Transcendental Extensions: Transcendental extensions. Transcendence bases theorem.

Ring extensions: Integral extensions. Noether Normalization Theorem. Integral closure. Integral Galois extensions. Going-up theorem. Decomposition groups. Extension of homomorphisms. Valuation rings.

Complements: Introduction to local fields. Dedekind domains. DVR's.


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Grades: Weekly homework 3/5, final 2/5.

Homework Policy: You can work together but you must write up your own solution in your own words, rather than copy a classmate's solution. You can refer to a result in a book (or any other source) ONLY if the result has been proved in the class. If a solution to a problem is given in a book (or any other source), you are allowed to use it, but you should write the solution in your own words; "copy-pasting" or referring to that source is not an acceptable solution. Homework is due on Mondays at 5 pm.

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