Problem Sheet 31
Polynomials over Rings, The division algorithm and Remainder Theorem.
1. Prove that if R is an integral domain and p(x) and q(x) are in R[x] then:
deg (pq) = deg p + deg q
where deg a means the degree of the polynomials a(x) ∈ R[x].
2. F Use the factor theorem to show that x4 + [4] can factorize as lineal factors in Z5
3. Find all Zp , p prime in which x2 + [2] divides x5 − [10]x + [12].
4. Find all odd primes p for which x − [2] is a factor of x4 + x3 + x2 + x in Zp
5. Prove that if p is prime then each element of Zp is a root of xp − x and deduce the
Wilson’s congruence (p − 1)! ≡ −1 mod p.
Hint: U (Zp ) is a group under multiplication of order p − 1.
6. Use the Division algorithm for polynomials to divide:
(a) 3x3 − 2x2 + 4x − 3 between x2 + 3x + 3 and x3 − 1 between x + 2 in Q[x] .
√
√
(b) F x2 − 3 between 2x2 + 1 in R[x].
(c) F 4x2 + 2x + 1 and 3x + 4 in Z5 .
(d) Factorize x3 + x2 + x + 1 in Z2 in irreducible factors. (Hint: Search for roots).
7. Use the division algorithm for polynomials to divide x3 − 5x2 + 3x − 15 and x2 + 3
in Q[x] . What is their greatest common divisor?
8. Find u(x) and v(x) in Q[x] such that
(x2 + x + 1)u(x) + (x3 + x + 1)v(x) = 1.
9. F Two polynomials are said to be relatively prime in F [X] if their greatest common
divisor is eF (the unity of the field.) Are x3 + x + 1 and x2 + x + 1 relatively prime
in Z3 ?.
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Esther Vergara Diaz, [email protected], see also http://www.maths.tcd.ie/~evd
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