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Review for the second midterm, Math 427
Prof. Lerman
11/13/15
Any past quiz or homework problem is fair game for the test.
The second midterm will mostly cover ring theory. However, you should not forget about the class
equation, Cauchy’s theorem, p-subgroups and semi-direct products.
practice problems
1
Explain why Sn is a semidirect product of An and Z2 .
2 Suppose G is a finite group of order 20. Does it have to have an element of order 2? 5? 4?
Explain.
3 Suppose R is a commutative ring with 1, p ∈ R is irreducible and u ∈ R a unit. Prove that up
is irreducible.
4 Suppose φ : G → H and ψ : H → K are two ring homomorphisms. Prove that their composite
ψ ◦ φ : G → K is also a ring homomorphism.
5 Consider the ideal (x) in the ring of polynomials R[x, y] in two variables. Prove that the quotient ring R[x, y]/(x) is isomorphic to the ring R[y]. What does it tell you about the ideal (x)? Is
it a maximal ideal? Is it a prime ideal?
6 Recall that every ideal I ⊂ Z is principal. Consider the ideals 6Z and 8Z in the ring Z. Find
m, n, k ∈ Z so that
mZ = 8Z ∩ 6Z,
nZ = 8Z + 6Z,
kZ = (8Z)(6Z).
7 Give an example of a principal ideal that is not prime.
Give an example of prime ideal that’s not principal.
Give an example of a prime ideal that is not maximal.
8
Prove that m, n ∈ Z are relatively prime if and only if mZ + nZ = Z.
9 Let R be a commutative ring with 1. Suppose that R has no proper ideals, that is, the only
ideals in R are 0 and R. Prove that R is a field. Hint: what’s R/0?
10
Let R be a ring with a subring A and let I ⊂ R be an ideal. Prove that
1. A + I := {a + i ∈ R | a ∈ A, i ∈ I} is a subring of R.
2. A ∩ I is an ideal in A.
3. I is an ideal in A + I.
4. (A + I)/I is isomorphic to A/(A ∩ I).
11 Let R be a ring, K, I ⊂ R ideals with K ⊂ I. Prove that I/K := {i + K | i ∈ I} is an ideal
in R/K and that (R/K)/(I/K) is isomorphic to R/I.
12
Let F be a field and p(x) ∈ F [x] be a polynomial of degree 3. Prove that p is irreducible if
and only if p has no roots in F .
13
Suppose a ring R is a UFD and {ai }∞
i=1 is a sequence of elements in R with (ai ) ⊂ (ai+1 )
for all i. Prove that the increasing chain of ideals (a1 ) ⊂ (a2 ) ⊂ · · · eventually stabilizes. That is,
show that there is m ∈ N so that (am+k ) = (am ) for all k ∈ N. Hint: write a1 as a product of
irreducibles/primes. (a1 ) ⊂ (an ) means that an |a1 . Therefore ...
14
Prove that R[x]/(x2 + 2) is isomorphic to the field of complex numbers.