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AbstractAlgebra

# AbstractAlgebra - Abstract Algebra Maths 113 Alexander...

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Unformatted text preview: Abstract Algebra Maths 113 Alexander Paulin December 8, 2010 Lecture 1 The basics: e-mail: [email protected]ath.berkeley.edu My website: math.berkeley.edu/ ∼ apaulin/ I’m Scottish ⇒ Math = Maths. Office hours: Monday 11am -12.30pm and Wednesday 11am - 12.30pm Grading: 1 Homework per week : Total worth 10 percent 2 midterms each worth 15 percent 1 Final Exam worth 60 percent Make sure now that you can make the final - it’s on the 16th of December at 3pm. If you can’t then you can’t take the course. We won’t be using a specific text. I’ll give out course notes each lecture. Books you may want to look at include: Classic Algebra by P.M.Cohn - This is hard but is the gold standard in my opinion. Algebra by Michael Artin - fantastic and easier to digest. Frankly there are loads of good basic books. These notes will be enough to understand everything, but going to the library to investigate concepts in more depth is a very good skill to learn. 1 What is Algebra If you ask someone on the street this question the most likely response is ”something horrible to do with Xs and Y s ”. If you’re lucky enough to bump into a mathematician then you might get something along the lines of ”Algebra is the abstract encapsulation of our intuition for how arithmetic behaves”. Algebra is deep . Deep means that it permeates all of our mathematical intuitions. In fact the first mathematical concepts we ever encounter are the foundation of the subject. Let me summarize the first six years of your mathematical education: The concept of unity: The number 1. You always understood this, even as a little baby! ⇓ N := { 1 , 2 , 3 ... } . The natural numbers. Comes equipped with two natural operations + and × . ↓ Z := { ...- 2 ,- 1 , , 1 , 2 , ... } . The integers. We form these by using geometric intuition thinking of N as sitting on a line. Z also comes with + and × . Addition is particularly nice on Z , e.g. we have additive inverses. ↓ Q := { a b | a, b ∈ Z , b = 0 } . The rationals. We form these by formally dividing through by non-negative integers. We again use geometric insight picture Q . Q comes with + and × . This time multiplication is nice on Q \{ } , e.g multiplicative inverses exist. Notice that at each stage the operations of + and × become better behaved. These ideas are very simple, but also profound. We spend years understanding how + and × behave on these objects. e.g. a + b = b + a ∀ a, b ∈ Z , or a × ( b + c ) = a × b + a × c ∀ a, b, c ∈ Z . The central idea behind modern Algebra is to define a larger class of objects (sets with extra structure), of which Z and Q are canonical members. Canonical means definitive. ( Z , +) ⇒ Groups ( Z , + , × ) ⇒ Rings ( Q , + , × ) ⇒ Fields 2 If you’ve done Linear algebra before the analogous idea is ( R n , +) ⇒ V ector Spaces over R The amazing thing is that these vague ideas mean something very precise and have far far more depth than one could ever imagine....
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