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.