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September Learn how and when to remove this template message. Fundamental mathematical theorems.
They do stuff. Elements, such as X , which are not algebraic are called transcendental. In model theory , a branch of mathematical logic , two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. We then developed the Galois theory of finite fields, which turned out to be quite simple once we came up with the concept of the Frobenius automorphism. Lecture 1 Jan 31 We discussed how to define numbers. October
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You should write up your own solutions--direct copying is unacceptable. As a rough guideline for writing up solutions to homework problems, you should include enough detail so that i you can convince me that you understand the solution and ii you can understand your solution when you study for the final exam. Two assignments will be given under the rules for a takehome exam--no consultation with other students. You don't need to know Zorn's lemma this semester.
There are many good introductory books on Galois Theory, some of which are listed in the In these notes, a ring will always be a unital ring, i.e., a ring with unity 1 = 0. . Of course, we can view Z as a subring of any subfield of the complex. P. Remark. As every field is an ID the last proposition holds in particular for fields. Definition. Let K be a field. The prime subfield of K is the intersection of all the.
The rest of the material should be familiar if you had Math at NIU. Here is the table of contents:.
Hungerford, Algebra This is an excellent reference book, and has been a standard since it was originally published in The down side is that students seem to find it difficult to learn material here for the first time. Many of the proofs say "sketch of proof" and you have to work to figure out what was left out. Dummit and Foote, Abstract Algebra This book has a wealth of examples, and so it is an excellent reference book.
In I used it for the first half of , and then found it did not fit our syllabus for the second half. My one criticism is that so much material is included that it seems to be hard for students to decide what is really important. I would also like the first part to be written a little more tightly, like other graduate textbooks.