FEBRUARY ’10 TBL MATH 28A — LECTURE 13 — OUTLINE NOTES

We can now finish the job with unique factorization! Theorem 14. In any Euclidean domain R, every element can be factorized uniquely into irreducible elements. Proof. We know that every element can be factored into irreducible elements; the only point at issue is uniqueness. We will show by induction on N that all elements of size N have a unique factorization. Suppose that n ∈ R can be factored into irreducibles in two different ways

n = p1 . . . pk = q1 . . . ql we must show that k = l and, after reordering, and replacing elements with associates, p1 = q1, p2 = q2, etc. Since p1 is irreducible, it is prime, and since it divides q1 . . . ql, it must divide one of the qi. Re- ordering, we may suppose it divides q1. Then since q1 is prime, p1 and q1 must be assosciates. Since we are allowed to replace the qi with assosciates, we may suppose q1 = p1. Then dividing,

p2 . . . pk = q2 . . . ql and both sides have smaller size than N. Thus we inductively know k = l and (after reordering and replacing with associates) p2 = q2, . . . , pk = ql. Thus we are done. §4. Vector spaces— So far, we have been studying generalizations of rather familiar objects in math. The next kind of object we’re going to study is meant to generalize a situation that’s perhaps less familiar, and so it’s very important that we look at an example first to get on the same page. The key example we will be looking at the following: ♦ the collection of all polynomials p with real coefficients of degree ≤ 3 which satisfy p(1) = 0. Let us ask ourselves how such polynomials behave. We will make a series of observations. 1. We can certainly add such polynomials together, and the addition will respect the same associative and commutative properties that addition has in rings and fields. 2. It is therefore perhaps natural to ask ourselves which of the other properties of addition in rings will hold here. There were two: the existence of an additive identity, and the existence of additive inverses. There is certainly an additive identity: the zero polynomial. Also, there are inverses; we can take the negative such a polynomial and it is still a polynomial. Thus all four of the addition axioms for a ring are satisfied here too. 3. The situation for multiplication, however, is much less clear cut. We can’t really multiply elements of the collection ♦ together in general and expect to get another member of the collection. And though there are a few special cases where multiplication of elements does work, this turn out to be not a terribly fertile lead to pursue. So we say we can’t usefully multiply an element of the collection ♦ with each other. 4. Is there no role for multiplication here at all? No. There is some sense in which these elements can be multiplied—not with each other, but with real numbers. We can always multiply any element of the collection by any real number. 5. What properties does this multiplication have? Well, we do have that for any two real numbers a, b and any polynomial p from our collection, (ab) · p = a · (b · p). This is called compatibility of multiplication with real multiplication. It’s also the case that 1 · p = p. 6. Are there any other nice properties that we could hope to think about? In rings, we had one more kind of property—the ‘distributive’ one that includes addition and multiplication together. There are also distributive properties for the collection ♦. In fact there are two, one coming from addition in the reals and one coming from addition in the collection. So if a, b are reals and p, q are polynomials in our collection, (a + b) · p = a · p + bp,˙ and a · (p + q) = a · p + a · q.