Vol. 10 No 3 - Pi Mu Epsilon

VARBALOW AND VELLA, FACTORIZATION OF POLYNOMIALS 199it is only carried out for n = 2 and n = 3. When this is attempted forlarger values of n, partitions enter the picture. Indeed, if p(X) is a monicpolynomial of degree n, it factors uniquely as a product of monic irreduciblepolynomials p(X) = II pi (X) . So if we let di = deg(pi (X)) , then (2)implies that the set of degrees { dl, d2, - , dm } is a partition of n.Furthermore, p(X) is itself irreducible only if the partition so obtained is [n],otherwise each di < n. A typical monic polynomial of degree n has theformthe binomial coefficient as rather than [p:p2 ] or (;) .Derived partitions for PAwith the ai's in F. Let F be finite from now on, with q = \ F 1 . Nowthere are exactly q choices for each of the n ai's, so there are q n monkpolynomials of degree n altogether. Our goal is to count the number ofreducible polynomials of degree n (provided Nc (di) for di < n has alreadybeen computed), and subtract this number from q n to find Np(di).First we consider some preliminary information regarding partitions.We remind the reader that the conventions about 0 from Section I are stillin effect. We now introduce an operation on P. Given a partition IT I- n,recall that the length I(v) is the total number of (nonzero) parts and thedepth d(v) is the number of distinct (nonzero) parts. From (1) it followsthat the set of multiplicities S(v) = { 7r,, 7r2, - , vn } is a partition ofI (IT), which we call the derived partition of IT. Observe that 6 (v) haslength equal to the depth of IT (many of the multiplicities ui are 0). Forexample, if T = [1 3 , 2, 42 , 531, then v I- 28 and there are 9 parts, butonly 4 of them are distinct, so I (IT) = 9 and d(v) = 4. The multiplicitiesare {3, 1,0, 2, 3}, leading to S(v) = [ 1 , 2, 32], a partition of 9 of length4. We further illustrate this with the table on the next page.Next, suppose that v ? Pn; T = {p,, p2, -, pm} . Then we remindthe reader that the expression n! /II:, pi! is an integer, known as aInmultinomial coefficient, and often written as. We willPI 9 P29 .*'> Pmfurther abbreviate this to simply , writing the name of the partition on(:)the bottom. When m = 2, we will follow the usual convention of writingWe are now prepared to prove the following:THEOREM 1 : Let F be a finite field with 1 FI = q. Then the totalnumber of monic polynomials of degree n with coefficients in F is given by:Proof: We have observed above that the left side of (3) is correct. Wenow count the monic polynomials a different way to see that the right sideis also correct. Let p(X) be one, and let p(X) = II , pi (X) be itsfactorization into irreducible monic polynomials. Let v be the degree set{dl, d2, -, dm} (listed with multiplicities). Since F[Xl is a commutativering, the di's may be listed in any order, so as noted above, IT belongs topn.Let j be an integer with 1 < j S n. So there are TT, of the irreduciblefactors pi(X) with degree j. If Nj represents the number of ways ofchoosing these factors of degree j, then the multiplication principle ofcounting yields that there are Nj ways of choosing all the factorstogether, so this accounts for the product in (3).It remains to show that