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Math 331-2 Lecture Notes Math 331-2: Abstract Algebra Northwestern University, Lecture Notes Written by Santiago Ca~nez These are notes which provide a basic summary of each lecture for Math 331-2, the second quarter of \MENU: Abstract Algebra", taught by the author at Northwestern University. The book used as a reference is the 3rd edition of Abstract Algebra by Dummit and Foote. Watch out for typos! Comments and suggestions are welcome. Contents Lecture 1: Introduction to Rings 2 Lecture 2: Matrix and Polynomial Rings 6 Lecture 3: More Ring Constructions 8 Lecture 4: Quotient Rings 14 Lecture 5: More on Quotients and Ideals 18 Lecture 6: Maximal Ideals 21 Lecture 7: Prime Ideals 25 Lecture 8: Geometry via Algebra 28 Lecture 9: Rings of Fractions 32 Lecture 10: More on Fractions 37 Lecture 11: Chinese Remainder Theorem 43 Lecture 12: Euclidean Domains 47 Lecture 13: More on Euclidean 51 Lecture 14: Principal Ideal Domains 54 Lecture 15: Unique Factorization 57 Lecture 16: More on UFDs 60 Lecture 17: Polynomials Revisited 63 Lecture 18: Gauss's Lemma 66 Lecture 19: Eisenstein's Criterion 69 Lecture 20: Modules over Rings 72 Lecture 21: Module Homomorphisms 77 Lecture 22: Sums and Free Modules 80 Lecture 23: Modules over PIDs 85 Lecture 24: Structure Theorems 89 Lecture 25: More on Structure Theorems 92 Lecture 26: Jordan Normal Form 97 Lecture 27: More on Normal Forms 103 Lecture 1: Introduction to Rings We continue our study of abstract algebra, this quarter focusing on rings and (to a lesser extent) modules. Rings provide a general setting in which \arithmetic" and the notion of \number" makes sense, and indeed the development of ring theory was borne out of attempts to understand general properties analogous to those of integers. Last quarter we first motivated the study of groups by hinting at their use in describing permutations of roots of polynomials (and in the problem of deciding when a formula for such roots could be found), and rings will now provide the language with which we can begin to discuss these roots themselves. (The full story of these roots belongs to the study of fields, which is are special types of rings we will look are more carefully next quarter.) The theory of modules generalizes the subject of linear algebra, and for us will really put to use next quarter when we consider the relation between fields. Historically, the development of ring theory really took off in the mid-to-late 19th century in the following way. You may have heard of Fermat's Last Theorem, which is the result that for n > 2 there are no non-trivial solutions to xn + yn = zn: (Solutions such that x = 1; y = 0; z = 1 and so on are the trivial ones.) This problem has a very rich history, and attempts to provide a valid proof led to much new mathematics. (The final proof was only given a few decades ago, and relies on some very heavy machinery in number theory and algebraic geometry.) One early attempt towards a proof came from trying to exploit a certain factorization: rewrite the given equation as xn = zn − yn and factor the right side: xn = zn − yn = (z − y)(something)(something) ··· (something) using something analogous to z2 − y2 = (z − y)(z + y) in the n = 2 case. The individual factors on the right involve more than just integers, but we are not interested in their precise form here. Then, by trying to \factor" these terms on the right further we can try to come up with something like a \prime factorization" for the xn on the left, only where we have to be more careful here about what exactly counts as a \prime" in this context. If you could somehow show that there could not be enough \primes" on the right to give an n-th power xn overall, you would have a proof of Fermat's Last Theorem, and indeed this is what many people in the 19th century thought they had done. (If Fermat did actually have a proof of this result himself, it is very likely that it was phrased along these lines.) But, it was soon realized that such an argument relied on the uniqueness of \prime" factoriza- tions. For instance, in the case of integers we know that 243572 is not the square of an integer since the 35 term is not a square and there are no other primes apart from 2; 3; 7 we could use to express this number in an alternate way as a potential square, due to the uniqueness of the primes 2; 3; 7 showing up in this expression. If, however, such a factorization were not unique, we would not be able to derive information about a specific number (such as it not being a square) based solely on the form of one prime factorization alone. This causes a problem for the approach to Fermat's Last Theorem outlined above, since it depends on conclusions made from a single \prime" factorization, and in general in the types of sets of \numbers" (i.e. rings) which show up here, there is no reason to expect that such a uniqueness must hold. To give an explicit example of where a \prime" factorization can fail to be unique, consider the following set of complex numbers: p p Z[ −5] := fa + b −5 j a; b 2 Zg: 2 p (The notation on the left is pronounced \Z adjoin −5". We will see what this notation really means soon enough, but it basicallyp describes the structure obtained by taking all possible sums and products of integersp and −5 and things built up from these.)p Note that adding or multiplying any two elements of Z[ −5] produces an elementp still within Z[ −5], which is essentially why this is a \ring". In this ring, we can factor 6 2 Z[ −5] in the following two ways: p p 6 = 2 · 3 = (1 + −5)(1 − −5): We claim that both of these are \prime" factorizations of 6! Actually, we should now be more careful with terminology: what really matters is that these are factorizations into irreducible elements, where an \irreducible" element is essentially one which cannot be factored any further. (This sounds pretty much like the ordinary definition of \prime" with which you're probably familiar in the context of integers, but we will see later that the term \prime" actually means something different in ring theory; the notions of \prime"p and \irreducible"p are the same in Z and many other settings, but not always. Showing that 1+ −5 and 1− −5 are irreducible isp a simple brute-force check we will come back to later.) Thus, the factorization of elements of Z[ −5] into irreducible elements is not unique, so care has to be taken when working with this ring. Understanding this failure to have unique factorization in certain contexts played an important role in the development of ring theory. It turns out that this lack of uniqueness can be salvaged in many (not all) cases by considering factorizations of what were historically called \ideal numbers", and what we now call ideals, instead. The factorization of ideals is not something we will study in this course, since it belongs more properly to a course in (algebraic) number theory, but ideals in general are of central importance in ring theory and this is the setting in which they were first introduced. For our purposes, the point will be that ideals play a role in ring theory analogous to the one played by normal subgroups in group theory. Rings. We now come to formally defining the notion of a ring. A ring will have two operations, as opposed to a group which only had one, but the simplest way to start is with a group itself. Indeed, a ring is an abelian group (R; +)|where we think of the group operation + as \addition"| equipped with a second binary operation · : R × R ! R (which we think of as \multiplication", and usually write simply ab instead of a · b) satisfying: • associativity: (ab)c = a(bc) for all a; b; c 2 R, and • distributivity: a(b + c) = ab + ac and (b + c)a = ba + ca for all a; b; c 2 R. The distributivity requirement is important since it is the one which expresses a compatibility between the two ring operations|without it, addition and multiplication would just be two random operations with no relation to each other, and that's no fun. Note that both orders a(b + c) and (b + c)a have to be given explicitly in the definition since the ring multiplication is not assumed to be commutative. Addition is assumed to be commutative, simply because every type of \addition" we've ever come across is commutative, which is not the case for \multiplication". (In fact, as the book suggests, that + is commutative can be derived from the ring axioms alone, at least in the case where the ring has a multiplicative identity. We will define this formally in a bit.) Examples. Many examples of groups we saw last quarter can also be viewed as rings. For instance, Z, Q, and R under their usualp operations. Also, Z=nZ is a (finite) ringp under addition and multiplication mod n. The set Z[ −5] of complex numbers of the form a + b −5 with a; b 2 Z we mentioned in our introductory discussion is also a ring, and indeed it is a subring of C, meaning that it is a subgroup with respect to addition which is closed under multiplication.
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