Mathematics 228(Q1), Assignment 6
Due : Friday, March 16, 2007
Exercise 1.(20 marks) Let Rbe a ring. Which of the following subsets of R[X] are subrings ? Be sure to
justify your answer.
(a) All polynomials with constant term 0.
(b) All polynomials of degree ≤k,ka fixed positive integer.
(c) All polynomials in which the odd powers of Xhave zero coefficient.
Exercise 2.(20 marks) Find the monic greatest common divisor d(X) of the polynomials f(X) and g(X),
and polynomials s(X) and t(X) such that
d(X) = s(X)f(X) + t(X)g(X),
in the following cases.
(a) f(X) = X5+ 1 and g(X) = X2+X+ 1 in Q[X].
(b) f(X) = X6+X3+ 1 and g(X) = X3+X+ 1 in Z2[X].
(c) f(X) = 2X4+X2−X+ 1 and g(X) = 2X−1inZ5[X].
(d) f(X) = X+a+band g(X) = X3−3abX +a3+b3in Q[X]. (Here, aand bare arbitrary rational
numbers.)
Exercise 3.(10 marks) Let p > 0 be a prime number.
(a) By counting products of the form (X+a)(X+b), show that there are exactly (p2+p)/2 monic polynomials
of degree 2 that are not irreducible in Zp.
(b) Show that there are exactly (p2−p)/2 monic irreducible polynomials of degree 2 in Zp[x]. (Hint : Count
the number of monic polynomials first, and then use (a)
Exercise 4.(10 marks) Show that X2+ 1 is reducible in Zpif and only if there exists integers aand bsuch
that p=a+band ab ≡1 mod p.
Exercise 5.(10 marks) (a) Find the remainder when f(X) is divided by g(X) in the following cases.
(i) f(X) = 2X5−3X4+x3−2X2+X−8 and g(X) = X−10 in Q[X].
(ii) f(X) = 2X5−3X4+X3+ 2X+ 3 and g(X) = X−3inZ5[X].
(b) Determine if h(X) is a factor of f(X) in the following cases.
(i) h(X) = X+ 2 and f(X) = X3−3X2−4X−12 in R[X].
(ii) h(X) = X−2 and f(X) = X6−X3+X−5 in Z7[X]