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Mathematics 676-4 28) Homework (due Mar. 19) A. Hulpke Factorize the polynomial x15 + 2x13 + 4x12 + 16x11 + 136x9 + 64x8 − 240x7 + 416x6 + 832x5 −1792x4 + 5072x3 + 6912x2 + 3232x + 6464 using the Hensel-lifting method. 29) Let f ∈ Z[x] be a polynomial with leading coefficient c and n = deg( f ). Show that g := cn−1 f (x/c) ∈ Z[x] is monic and that the irreducible factors of f are in one-to-one correspondence with the irreducible factors of g. (Thus we may assume that a polynomial f ∈ Z[x] to be factored is monic.) 29) Let F be a perfect field (i.e. a finite field or a field in characteristic zero), f (x) ∈ F[x] irreducible, f (θ) = 0 and the norm function N: K[x] → F[x] as defined in problem 8. Let g ∈ F(θ)[x]. a) Show that there is an n ∈ Z such that N(g(x + nθ)) is squarefree. b) Now assume that N(g) is squarefree, and that g = ∏ gi is the factorization in irreducible factors over F(θ). Show that N(g) = ∏ N(gi ) is the factorization of N(g) into irreducible factors over Q[x] and that gi = gcd(g, N(gi )). c) Based on this, describe a method for factorization of polynomials in F(θ). d) Let f (x) = x3 + x + 1 ∈ Q[x], θ a root of f and g(x) = x3 + (θ2 + θ + 1)x2 + (θ2 − 1)y − θ. Determine the irreducible factors of g over Q(θ). (You may use the computer to perform a factorization over Q, as well as to calculate gcd’s. In GAP, if g is a polynomial over the rationals, and y an indeterminate over an AlgebraicExtension, then Value(g,y) creates the corresponding polynomial over the extension. The conversion vice versa would be much more technical and the easiest solution will be to retype a polynomial defined over the extension to create the corresponding rational polynomial.) 31∗ ) Let F be a field and x1 , . . . , xn be indeterminates. A polynomial f ∈ F[x1 , . . . , xn ] is called symmetric, if for any permutation σ of the indeterminates we have that σ( f ) = f . Obviously,the following functions (called elementary symmetric polynomials) are symmetric: S1 (x1 , . . . , xn ) = x1 + x2 + x3 + · · · + xn S2 (x1 , . . . , xn ) = x1 x2 + x1 x3 + · · · + xn−1 xn S3 (x1 , . . . , xn ) = ∑ x1 x j xk i< j<k .. . Sn (x1 , . . . , xn ) = x1 x2 · · · · · xn The aim of this exercise is to prove the fundamental theorem of elementary symmetric polynomials: Theorem: Let f ∈ F[x1 , . . . , xn ] symmetric, then exists g ∈ F[x1 , . . . , xn ] such that f (x1 , . . . , xn ) = g S1 (x1 , . . . , xn ), S2 (x1 , . . . , xn ), . . . , Sn (x1 , . . . , xn ) . Note: We have been using this theorem in the proof of the Frobenius/Chebotarev theorem. Example: x12 x22 x3 + x12 x2 x32 + x1 x22 x32 + x12 + 2x1 x2 + 2x1 x3 + x22 + 2x2 x3 + x32 = S12 + S2 ∗ S3 . We define an ordering on monomials by setting: x1d1 x2d2 · · · xndn > ye11 ye22 · · · yenn if and only if for some i: d j = e j for j < i and di > ei (lexicographic comparison of the exponnet vectors). a) Suppose that d1 ≥ d2 ≥ · · · ≥ dn is a sequence of integers. Show that x1d1 x2d2 · · · xndn is the largest (with respect to this ordering) term of the polynomial n−2 Td1 ,...,dn (x1 , . . . , xn ) := S1d1 −d2 −d3 −···−dn · · · · · Sn−2 d −dn−1 −dn d n−1 · Sn−1 −dn d n−1 · Sn−1 −dn · Sndn b) Suppose that f (x1 , . . . , xn ) is symmetric and c · x1d1 x2d2 · · · xndn is the largest (with respect to this ordering) term of f and let Td1 ,...,dn (x1 , . . . , xn ) for these di as in a). Show that the largest monomial of f (x1 , . . . , xn ) − c · Td1 ,...,dn (x1 , . . . , xn ) is strictly smaller than x1d1 x2d2 · · · xndn . c) Show that by iterating the process in b) you can build the polynomial g as claimed in the theorem. d) Express f = x2 y + x2 z + 3xyz + y2 x + y2 z + z2 x + yz2 − 2x3 y3 z −4x3 y2 z2 − 4x2 y3 z2 − 2x3 z3 y − 4x2 y2 z3 − 2xy3 z3 as a polynomial in the elementary symmetric polynomials in x, yandz. Problems marked with ∗ are bonus problems for extra credit. I will be out of town from 3/6 until 3/13, as well as between 3/22 and 3/29.