1University of Illinois
MATH 417 Introduction to Abstract Algebra
Worksheet 3
1. (True/False) For each item below, indicate whether the statement is true or false.
No justification needed.
(a) (True/False) Let ϕ : R→ S be a ring morphism. If I is an ideal of R then ϕ(I)
is an ideal of S.
(b) (True/False) Every quotient of an integral domain is an integral domain.
(c) (True/False) If F is a field, then every proper nontrivial prime ideal of F [x] is
maximal.
(d) (True/False) If a group action is transitive, then it is faithful.
(e) (True/False) Let X be a G-set. The isotropy subgroup Gx is normal in G for
all x ∈ X.
2. Complete the following definitions:
(a) A group (G, ·) is a set G with binary operation · such that . . .
(b) A ring (R,+, ·) is a set R with binary operations +, · such that . . .
(c) A field F is . . .
(d) Let F be a field. A non-constant polynomial f ∈ F [x] is irreducible over F
if . . .
(e) Let R be a non-trivial commutative ring with unity. A prime ideal P is . . .
(f) Let R be a non-trivial commutative ring with unity. A maximal ideal M is
. . .
3. State the Fundamental Homomorphism Theorem for groups.
4. State the Division Algorithm for polynomials in F [x] with F a field.
25. State the Eisenstein Criterion.
6. State the Orbit-Stabilizer Theorem.
7. Find the subgroup diagram for Z12.
8. Find all abelian subgroups (up to isomorphism) of order 23 · 33.
9. For G = Z3×Z6 and H the cyclic subgroup generated by ([1], [1]) find each H-coset
in G.
10. Solve the congruence 12x ≡ 6 modulo 18.
11. Is the polynomial x4 + 3x3 + 6x2 − 6 ∈ Z[x] irreducible over Q? Why or why not?
12. Find all c ∈ Z3 such that Z3[x]/(x3 + cx2 + 1) is a field.
13. Partition the dihedral group D4 into its conjugacy classes.
14. Let ϕ : R→ S be a ring morphism.
(a) If J is an ideal of S show that ϕ−1(J) is an ideal of R.
(b) Suppose R, S are non-trivial commutative rings with unity. Let M be a max-
imal ideal of S. Is ϕ−1(M) a maximal ideal of R? Provide a proof or give a
counter-example.
(c) Suppose R, S are non-trivial commutative rings with unity. If ϕ is surjective
and P a prime ideal of S show that ϕ−1(P ) is a prime ideal of R.