Ring Theory (Math 113), Summer 2014

1 ring Theory (Math 113), Summer 2014 James McIvorUniversity of California, BerkeleyAugust 3, 2014 AbstractThese are some informal notes on rings and fields, used to teach Math 113 at UC Berkeley, Summer 2014. We go through the basic stuff: rings, homomorphisms, isomorphisms, ideals andquotient rings, division and (ir)reducibility, all heavy on the examples, mostly polynomial rings andtheir quotients. Some allusions to basic ideas from algebraic geometry are made along the we get into fields, culminating in a brief exposure to the basic ideas of galois Basic Examples and Examples .. of a ring .. elements in a ring ..52 Subrings; ; Adjoining Elements .. of Rings ..103 Kernels and kernel of a homomorphism .. on Ideals ..144 Quotient : Cosets.. Rings .. easy way to think about quotients of a polynomial ring .. Isomorphism Theorem ..175 Factorization; Reducibility; and Factorization inZ.

2 And Factorization of Polynomials ..196 Special Classes of .. Domains ( Domains ) .. Ideal Domains .. Domains .. Factorization Domains .. Between the Types of Rings ..2517 Prime and Maximal numbers and prime ideals inZ.. Ideals .. Ideals .. between these ideals and their quotients ..288 Field Extensions and Examples .. vs. Transcendental Extensions .. Extensions ..329 Minimal Polynomials of Finite Algebraic the Minimal Polynomial.. of the Primitive Element ..3510 Field Automorphisms and The Galois Automorphisms and Galois Groups .. Embeddings of Subfields.. Proof of Primitive Element Theorem ..3811 The Galois The Galois Connection .. Normality ..4112 Fundamental Theorem of Galois The Theorem .. A Burly Example..4413 Application: Solution by Radicals; Insolubility of the Relation Between Intermediate Subfields and Solution by Radicals .. Cyclic Galois Groups.

3 Insolubility of the Quinitc ..4521 Basic Examples and DefinitionsNext ExamplesA ring is just a set where you can add, subtract, and multiply. In some rings you can divide, and inothers you can t. There are many familiar examples of rings, the main ones falling into two camps: number systems and functions . : the integers .. , 2, 1, 0, 1, 2, .., with usual addition and multiplication, form a ring . Notethat we cannot always divide, since 1/2 is no longer an Similarly, the familiar number systemsQ,R, andCare all 2Z: the even integers .. , 4, 2, 0, 2, 4, .. [x]: this is the set of polynomials whose coefficients are integers. It is an extension ofZin thesense that we allow all the integers, plus an extra symbol x, which we are allowed to multiplyand add, giving rise tox2,x3, etc., as well as 2x, 3x, etc. Adding up various combinations of thesegives all the possible integer [x,y,z]: polynomials in three variables with integer coefficients.

4 This is an extension of theprevious ring , too. In fact you can continue adding variables to get larger and larger : The integers modn. These are equivalence classes of the integers under the equivalencerelation congruence modn . If we just think about addition (and subtraction), this is exactly thecyclic group of ordern, as discussed a long time ago. However, when we call it a ring , it meanswe are also using the operation of [0, 1]: This is my notation for the set of all continuous real-valued functions on the interval[0, 1]. For example,f(x) = 2xandg(x) = sinxare inC[0, 1]. They can be added and multipliedto give (f+g)(x) = 2x+ sinxand (fg)(x) = 2xsinx, which are also elements ofC[0, 1]. Thisis a very large ring , since there are lots and lots of continuous functions. Notice also that thepolynomials from example 2 are contained as a proper subset of this ring . We will see in a bitthat they form a subring.

5 (R) (non-commutative): the set ofn nmatrices with entries inR. These form a ring , sincewe can add, subtract, and multiply square matrices. This is the first example we ve seen wherethe order of multiplication matters:ABis not always equal toBA(usually it s not). [[x]]: this ring consists of what are called formal power series with entries inQ(the rationalnumbers). A power series is just a polynomial with (possibly) infinitely many terms, such as yousee in a calculus course. The word formal means that we don t care whether they convergeor not, so that the series n!xnis perfectly good, even though you never talk about it in calcu-lus because it only converges whenx= 0. Because of this possible non-convergence, we can tthink of these power series as functions, and we think of thexas a formal variable , ratherthan something for which we can substitute a numerical value.

6 We are restricting the coefficientsto be rational numbers for the sake of example, but you could just as well considerZ[[x]] orR[[x]].1In fact they refields, to be defined [{1p}pis prime]: We take the integers, and adjoin all fractions of the form1p, for each primenumberp. But since we can multiply elements in a ring , we can also obtain such fractions as16=12 13. Since we can add, we can obtain, for instance,56=16+16+16+16+16. So you getmore fractions than those with just primes in the denominator, and ones in the numerator. Whichfractions do we get? [x]/(x2+ 1). Here s some new notation. It means take the polynomial ringR[x] as above, and divide out by the polynomialx2+1, meaning that this polynomial gets set to zero. So in this ring ,the polynomial (x+1)2is the same as 2x, since (x+1)2=x2+2x+1 = 2x+(x2+1) = 2x+0 = way of thinking about this is thatx2is the same as 1. So there are never any powersofxlarger than 1, since whenever we get tox2we just swap it out for 1.

7 So every polynomialin here is going to have a constant term and anxterm and that s it. This should remind youof the complex numbers, which each have a real part (the constant term) and an imaginary part(thexterm), but usually when we work with complex numbers, we use the letteriinstead it s essentially the same ring . Note that for complex numbers, we can always divide (exceptby zero, of course), so that shows that in this weird polynomial ring , we can divide as well, whichis a bit strange, since in the usual polynomial ring we can almost never divide (since, for example,1xdoesn t count as a polynomial). This is an example of a quotient ring , which is the ring versionof a quotient group, and which is a very very important and useful Here s a really strange example. Consider a setS(finite or infinite), and letRbe the set of allsubsets ofS. We can makeRinto a ring by defining the addition and multiplication as two subsetsA,B, defineA+B=A B\A B(sometimes people call this the symmetricdifference, or exclusive or ).

8 Define subtraction by A=S\A(the set-theoretic complement).ThusA B= (A (S\B))\(A S\B). This example shows you that addition and multiplicationneedn t be the usual operations we know from grade school. But luckily, in most of our examples,like above, they will of a RingAs the preceding examples indicate, a ring is basically a set in which we have a way of adding, sub-tracting, multiplying, but not necessarily dividing2Of course, depending on the ring , the addition andmultiplication may not seem like the ordinary operations we are used to. So here s the formal definition:Definition a setRendowed with two binary operations, usually denoted + and , suchthat R1:Ris an abelian group with respect to + R2: For anya,b,cinR,a (b c) = (a b) c(associativity of ) R3: For anya,b,cinR,a (b+c) =a b+a c(left-distributivity) R3 : For anya,b,cinR, (a+b) c=a c+b c(right-distributivity)Most often we will also impose some additional conditions on our rings, as follows: R4: There exists an element, denoted 1, which has the property thata 1 = 1 a=afor allainR(multiplicative identity) R5:a b=b afor alla,binR(commutativity of )Notice that sinceRforms an abelian group under +, the addition isalwayscommutative, andthat there is also an additive identity, which we will usually denote by 0.

9 So axioms 4 and 5 impose2We will see later that a ring in which we can always divide is called conditions on the multiplicative structure ofR. A ring satisfyingR4 is called aring with unity(or sometimes aunital ring ), where unity is just a fancy name for the multiplicative identity. A ringsatisfying R5 is called acommutative usual we use exponents to denote compounded multiplication; associativity guarantees that theusual rules for exponents apply. However, with rings (as opposed to multiplicative groups), we must usea little caution, sinceakmay not make sense fork<0, asais not guaranteed to have an most of the examples above it is easy to see what the additive and multiplicative identities are they for example 10?The axioms are just aminimallist of properties of the addition and multiplication. Others can bededuced from these, ,Lemma R be a ring , with additive and multiplicative identities 0 and 1, respectively.

10 Thenfor all a,b in R , = 0;2.( a)b=a( b) = (ab);3.( a)( b) =ab;4.(na)b=a(nb) =n(ab)for any n 4, note that n is not to be thought of as an element of R : the notation na just means a+ +a,where there are n copies of a in the Exercise2. To show that ( a)b= (ab) is to show that the element ( a)bis the additive inverse ofab; sowe add them together, and hope to get zero. So ( a)b+ab= (( a) +a)b= (0)b= 0 (by 1).The equality ofa( b) and (ab) is Exercise4. (na)b= (a+ +a)b= (ab+ +ab) =n(ab) =a(b+ +b) =a(nb)Example (Thezero ring ) The axiom R4 begs the question: can 0 and 1 be the same? The answeris yes, but in that case it turns out that there is only one element in our ring , which is 0 (which is equalto 1). We call this the zero ring , and sometimes write it just as 0. Here s the reason: suppose 1=0 ina ring , and now pick any elementrin this ring . Sincer= 1 r= 0 r= 0, we find that every element is elements in a ringHere we pick out some types of elements that can occur in rings:Definition an element of a ringR.