Comprehensive Exam, Summer 2003
for
MATH 788F & MATH 788G
1. Prove Perron’s theorem that if a0,a1, . . . , anare integers with an= 1,a06= 0, and
|an−1|>|an|+|an−2|+|an−3|+· · · +|a1|+|a0|,
then anxn+an−1xn−1+· · · +a1x+a0is irreducible. You may use Rouch´
e’s theorem as stated below, but
otherwise your argument should be self-contained.
Rouch´
e’s Theorem Let f(x)and g(x)be polynomials in C[x], and let
C={z∈C:|z|= 1}.
If the strict inequality |f(z) + g(z)|<|f(z)|+|g(z)|holds for each z∈ C, then f(x)and g(x)have
the same total number of zeroes (counted to their multiplicities) inside the circle C(i.e., in the interior
of the region bounded by C).
2. Determine whether each of the following is Eisenstein. Justify your answers.
(a) x2−3x−2(b) x3+ 4x2+x−1
3. Prove Kroncecker’s theorem that if f(x)is an irreducible monic polynomial with all of its roots on the unit circle
(i.e., all of its roots have absolute value 1), then f(x)is a cyclotomic polynomial.
4. Let nbe a positive integer, and set
Sn=f(x) =
n
X
j=0
εjxj:εj∈ {0,1}for each jand ε0= 1.
Let g(x)be an arbitrary polynomial in Z[x]. Prove there are ≤2n/(n+ 1) polynomials in Sndivisible by g(x)2.
5. (a) Prove that x255 −1factors modulo 2as a product of distinct irreducible polynomials each of degree ≤8.
(b) It follows from part (a) that each irreducible factor of x255 −1modulo 2has degree in S={1,2,3, . . . , 8}.
Which numbers in Sactually occur as the degree of some irreducible factor of x255 −1modulo 2? Justify your
answer.
6. (a) Prove that the non-reciprocal part of 1 + x+x2+x4+x8+x16 +x32 is irreducible.
(b) Prove that the non-reciprocal part of 1 + x+x2+x4+x8+· · · +x2nis irreducible for every integer n≥2.
7. We showed that if nis a positive integer and a1, a2, . . . , anare distinct integers, then
(x−a1)(x−a2)· · · (x−an)−1
is irreducible. Determine with justification whether or not there exists an Nsuch that if nis an integer ≥Nand
a1, a2, . . . , anare rational numbers, then (x−a1)(x−a2)· · · (x−an)−1is irreducible over the rationals.
