Math 103B
Test 2
100 pts
February 25, 2009
------------------------------------------------------------------------------------------------------Directions: Justify all answers and show all work.
Notation: Z is the ring of integers, Q is the field of rationals, Zp is the field
of p elements, F[x] is the ring of all polynomials in x with coefficients in a field F.
-------------------------------------------------------------------------------------------------------(1) Find the remainder when x51 is divided by x + 2
in Z5[x].
(12 pts)
-------------------------------------------------------------------------------------------------------(2) Let f(x) and g(x) be nonzero polynomials in Q[x].
Prove that the polynomial x2 f(x)3 - g(x)3 cannot possibly be the zero polynomial
in Q[x]. Hint: Consider the leading term.
(12 pts)
--------------------------------------------------------------------------------------------------------(3) Let I be a nonzero ideal of Q[x]. To prove Q[x] is a PID, one takes the monic
polynomial g(x) in I of smallest degree, and aims to show that when any polynomial
f(x) in I is divided by g(x), the remainder r(x) equals 0.
Use the division algorithm to show that r(x) = 0.
(15 pts)
--------------------------------------------------------------------------------------------------------(4) Show that each polynomial below is irreducible in Q[x].
(i) x3 + 98x2 + 87x + 11
(12 pts)
(ii) x8 + 1
(12 pts)
-------------------------------------------------------------------------------------------------------(5) Write down an irreducible polynomial of degree 999 in Q[x]. Justify. (10 pts)
--------------------------------------------------------------------------------------------------------(6) Let u denote the real zero of the polynomial x3 + x + 1. (Thus u is roughly equal
to -.6823, but that's not important right now.) Let D be the integral domain consisting
of all numbers f(u), where f(x) runs through all polynomials in Q[x]. (For example,
the number u9 - (4/3) u7 + 6 is an element of D.)
(i) True or False: The set D equals the set of all numbers of the form a + bu + cu2
with rational coefficients a, b, c. Justify your answer.
(12 pts)
(ii) Prove that D is a field. Justify every step.
Hint: A certain ring homomorphism from Q[x] onto D may be of use.
(15 pts)