Abstract Algebra Maths 113Alexander PaulinDecember 8, 2010Lecture 1The basics:e-mail: [email protected] website: math.berkeley.edu/∼apaulin/I’m Scottish ⇒ Math = Maths.Office hours: Monday 11am -12.30pm and Wednesday 11am - 12.30pmGrading:1 Homework per week : Total worth 10 percent2 midterms each worth 15 percent1 Final Exam worth 60 percentMake sure now that you can make the final - it’s on the 16th of December at 3pm. Ifyou can’t then you can’t take the course. We won’t be using a specific text. I’ll give outcourse 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 understandeverything, but going to the library to investigate concepts in more depth is a very goodskill to learn.1What is AlgebraIf you ask someone on the street this question the most likely response is ”something horribleto do with Xs and Ys”. If you’re lucky enough to bump into a mathematician then youmight get something along the lines of ”Algebra is the abstract encapsulation of our intuitionfor how arithmetic behaves”.Algebra is deep. Deep means that it permeates all of our mathematical intuitions. Infact 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, 0, 1, 2, ...}. The integers.We form these by using geometric intuition thinking of N as sitting on a line. Z also comeswith + and ×. Addition is particularly nice on Z, e.g. we have additive inverses.↓Q := {ab|a, b ∈ Z,b(=0}. The rationals.We form these by formally dividing through by non-negative integers. We again usegeometric insight picture Q. Q comes with + and ×. This time multiplication is nice onQ\{0}, e.g multiplicative inverses exist.Notice that at each stage the operations of + and × become better behaved. These ideasare very simple, but also profound. We spend years understanding how + and × behave onthese objects. e.g.a + b = b + a ∀a, b ∈ Z,ora × (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 extrastructure), of which Z and Q are canonical members. Canonical means definitive.(Z, +) ⇒ Groups(Z, +, ×) ⇒ Rings(Q, +, ×) ⇒ F ields2If you’ve done Linear algebra before the analogous idea is(Rn, +) ⇒ V ector Spaces over RThe amazing thing is that these vague ideas mean something very precise and have farfar more depth than one could ever imagine.Sets and MappingsA set is any collection of objects, e.g. six dogs, all the protons on Earth, every thoughtyou’ve ever had, N, Z, Q, R. The objects we will study in this course will all be sets.NotationLet S and T be two sets.1. S ⊂ T - S is contained in T. Note that they may be equal.2. S ∩ T - The intersection of S and T.3. S ∪ T - The union of S and T .4. S × T={(a, b)|a ∈ S, b ∈ T }. The product of S and T.5. ∅ - The empty set.Definition. We say that S and T are disjoint if S ∩ T = ∅.Definition. A map f (or function) from S to T is a rule sending elements of S to T.f : S → Tx → f(x)A map f : S → T can have various nice propertiesDefinition. 1. f : S → T is injective if f (x)=f (y) ⇒ x = y∀x, y ∈ S.2. f : S → T is surjective if given y ∈ T there exists x ∈ S such that f(x)=y.3. If f : S → T is both injective and surjective we say it is bijective. Intuitively thismeans there is a one to one correspondence between elements of S and T.Some examples of maps:1. S =T = N,f : N → Nx → x22. S= Z × Z, T = Z,f : Z × Z → Z(a, b) → a + b3Equivalence RelationsWithin a set it is sometimes natural to talk about different elements being related to eachother. For example, on Z we could have∀a, b ∈ Z,a∼ b ⇐⇒ a − b is divisible by 2.Definition. An equivalence relation on a set S is a subset U ⊂ S × S satisfying:1. (symmetric) (x, y) ∈ U ⇐⇒ (y, x) ∈ U.2. (reflexive) ∀x ∈ S, (x, x) ∈ U.3. (transitive) Given x, y, z ∈ S if (x, y) ∈ U and (y, z) ∈ U ⇒ (x, z) ∈ U.In more convenient notation we write x ∼ y to mean (x, y) ∈. An equivalence class isa maximal subset of elements of S which are equivalent to each other. This makes sensebecause of the above axioms. I leave it as an exercise to show that the union of all theequivalence class is equal to S and they are all disjoint. We say they form a partition of S.These are all the basic set theoretic concepts we’ll needBack to ZWe may naturally express + and × in the following set theoretic way:+:Z × Z → Z(a, b) → a + b× : Z × Z → Z(a, b) → a × bHere are 4 properties that + satisfies:1. (associativity): a +(b + c)=(a + b)+c ∀a, b, c ∈ Z2. (Existence of additive identity) a + 0 = 0 + a = a ∀a ∈ Z.3. (Existence of additive inverses) a +(−a)=(−a)+a =0∀a ∈ Z4. (Commutativity) a + b = b + a ∀a, b ∈ Z.Here are 3 properties that × satisfy:1. (associativity): a × (b × c ) = (a × b) × c ∀a, b, c ∈ Z2. (Existence of multiplicative identity) a × 1=1× a = a ∀a ∈ Z.3. (Commutativity) a × b = b × a ∀a, b ∈ Z.+ and × interact by the following:1. (Distributivity) a × (b + c) = (a × b)+(a × c) ∀a, b, c ∈ Z.4Remarks1. each of these properties is totally obvious but will for the foundations of future defini-tions: Groups and Rings.2. All of the above hold for Q. There is an extra property that non-zero elements havemultiplicative inverses e.gGiven a ∈ Q\{0}∃b ∈ Q such that a × b = b × a =1.This extra property will motivate the definition of a field.3. The significance of the associativity laws is that summing and multiplying a finitecollection of integers makes sense, i.e. is independent of how we do it.5Abstract Algebra Maths 113Alexander PaulinSeptember 2, 2009Lecture 2Last time we introduced the basic language of set theory we required. We then expressed+ and × on Z in this language. We wrote down eight elementary properties they satisfied.Let’s look at some of their deeper properties on Z. Again, these will motivate definitions ina more abstract setting.It is an important
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