Download Practice Problems for the Final Exam - MATH 4010

1 Abstract Algebra Professor M. Zuker
Practice Problems for the Final Exam - MATH 4010 May 2015
1. Quick problems.
(a) Let a, b and m be non-zero integers with m > 1. Let d = gcd(a, m) and suppose
that d|b. How many distinct solutions X, does the linear congruence aX ≡ b
mod m have modulo m? A. Can’t say with the information gives. B. Exactly
one. C. None. D. d E. m/d F. lcm(a, m).
(b) How many solutions does 65X ≡ 52 mod 169 have modulo 169?
(c) Let R = Z81 [x]. For any integer n ≥ 0, give an example of a polynomial of degree
n that is a unit.
(d) In a commutative ring with unity, let u1 and u2 be units. Show that u1 u2 6= 0.
Do the units form a group under multiplication?
√
√
(e) Show that R = Z( 5i) is an integral domain. ( R = {a+b 5i | a, b ∈ Z, i2 = −1}.
√
(f) Show that the only units in Z( 5) are ±1.
(g) Give an example of a prime ideal of Z × Z that is not maximal.
(h) Give an example of a maximal ideal of Z[x] that is not principal.
(i) Let f (x) = x30 − 12. Show that f (x) is irreducible over Q. What is the smallest
degree of an extension field E of Q such that f (x) factors? Define E explicitly.
(j) Let f (x) = x30 − 15. Show that f (x) is irreducible in Q[x]. What is the degree of
the splitting field of f (x) over Q. Hint: The splitting field is of the form Q(α, β).
2. Let R be a √
commutative ring with unity. For any ideal N , define the “radical” of N ,
denoted by N by:
√
N = {r ∈ R | rn ∈ N for some n ∈ Z+ }.
The radical of the trivial ideal, {0} is called the “nilradical”. Prove that the radical of
an ideal is itself an ideal.
p
3. Let R be a commutative ring with unity. Prove that the quotient ring R/ {0} (R
modulo the nilradical) has no nilpotent elements.
m
4. Prove that the polynomial p(x) = x7 − 2 is irreducible in Z29 [x]. Hint: Compute x29
modulo p(x) for 1 ≤ m < 7 and show that the result is never x.
5. Prove that x2 − 3 and x2 − 11 are both irreducible in Z31 [x]. Let α be a zero of x2 − 3
in some extension field and let β be a zero of x2 − 11 in some extension field. Show
that [Z31 (α, β) : Z13 ] = 2. Find a ∈ Z31 such that β = αa.
6. Suppose that p is a prime number and p ≡ 3 mod 4. Let a be a quadratic residue
modulo p. That is, a ∈ Zp and a = b2 for some b ∈ Zp . Show that
b = ±a
p+1
4
.
7. Show that xp − x + 1 is irreducible in Zp [x] where p is any prime number.