MA 405, Abstract Algebra, Section 1
Worcester State University, Spring 2018

Homework 10, due Monday, May 7th:

Do, but do not hand in:
1. Prove that if \([E:F]=p\) where \(p\) is prime, then the only subfields of \(E\) containing \(F\) are \(F\) and \(E\).

2. Determine all the subfields of:
(a) \(\mathbb{Q}(\sqrt{7})\)
(b) \(\mathbb{Q}(\sqrt[3]{5})\)
(c) \(\mathbb{Q}(\sqrt[5]{11})\)
(d) \(\mathbb{Q}(\sqrt{7}, \sqrt{13})\)

3. Prove that \(\mathbb{Q}(\sqrt{2} + \sqrt{3}) = \mathbb{Q}(\sqrt{2}, \sqrt{3})\).

Hand in:
1. Prove that \(\mathbb{Q}(\sqrt{-3})\) is the splitting field for \(x^3 - 1\) over \(\mathbb{Q}\).

2. Prove that \(\mathbb{Q}(\sqrt{7}, i)\) is the splitting field for \(x^4 - 6x^2 - 7\) over \(\mathbb{Q}\).

3. Let \(\theta\) be an element of an extension of a field \(F\), and let \(a, b \in F\) with \(a \neq 0\). Prove that \(F(\theta) = F(a \theta + b)\).

4. Let \(\theta = \sqrt{3} + \sqrt{-5} \in \mathbb{C}\).

(a) Prove that \([\mathbb{Q}(\theta):\mathbb{Q}] = 4\).

(b) Find the minimal polynomial for \(\theta\) over \(\mathbb{Q}\).

5. Let \(\alpha \in \mathbb{C}\) be algebraic over \(\mathbb{Q}\).
(a) Prove that \(\alpha^n\) is algebraic over \(\mathbb{Q}\) for any \(n \in \mathbb{Z}\).

[Hint: Find an algebraic extension of \(\mathbb{Q}\) containing \(\alpha^n\).]

(b) Prove that \(\alpha^{1/q}\) is algebraic over \(\mathbb{Q}\) for any positive integer \(q\).

[Hint: Use the minimal polynomial for \(\alpha\) over \(\mathbb{Q}\) to construct a polynomial having \(\alpha^{1/q}\) as a zero.]

(c) Prove that \(\alpha^r\) is algebraic over \(\mathbb{Q}\) for any \(r \in \mathbb{Q}\).

[Hint: Use parts (a) and (b).]

Please typeset in LaTeX your solution to Exercise 5. Include a statement of the problem. This exercise will be graded both on correctness and presentation.

Homework 9, due Friday, April 20th:

Do, but do not hand in:
1. Prove that if \(D\) is an integral domain, \(a \in D\) is irreducible, and \(u \in D\) is a unit, then \(au\) is irreducible.

Hand in:
1. Prove that \(\mathbb{Z}[\sqrt{-3}]\) is not a unique factorization domain.

[Hint: Use the properties of the norm \(N(a+b\sqrt{-3}) = |(a+b\sqrt{-3})(a-b\sqrt{-3})| = a^2 + 3b^2\).]

2. Let \(D\) be an integral domain, and let \(a,b \in D\). Prove that if \(a \in (b)\) and \(b \in (a)\), then \((a) = (b)\).

3. Let \(D\) be a principal ideal domain. Prove that an ideal in \(D\) is prime if and only if it's maximal.

[Hints: One direction is easy. For the other, observe that if \((p) \subseteq (q)\), then \(p = dq\) for some \(d \in D\). If \((p)\) is a prime ideal, \(p=dq \in (p)\) implies \(d \in (p)\) or \(q \in (p)\). Consider both cases and show that in one, \((q) = (p)\), and in the other, \((q) = D\).]

4. Give a construction of a finite field of order 16. List the elements of the field. Choose a nonzero, nonidentity element of the field and find its inverse.

Note: Either list the elements carefully, as cosets, or carefully explain whatever notation you use to simplify your description.

Please typeset in LaTeX your solution to Exercise 3. Include a statement of the problem. This exercise will be graded both on correctness and presentation.

Homework 8, due Wednesday, April 11th:

Hand in:
1. Let \(p(x) = x^3 + x^2 + x + 1\).
(a) Write \(p(x)\) as a product of irreducible polynomials over \(\mathbb{Q}\).

(b) Write \(p(x)\) as a product of irreducible polynomials over \(\mathbb{Z}/2\mathbb{Z}\).

2. Prove that \(x^2 - 5\) is irreducible over \(\mathbb{Q}\) but reducible over \(\mathbb{Q}[\sqrt{5}]\).

3. Prove that for every positive integer \(n\) there are infinitely many polynomials of degree \(n\) in \(\mathbb{Z}[x]\) that are irreducible over \(\mathbb{Q}\).

4. Let \(F\) be a field and let \(f(x) \in F[x]\) be reducible over \(F\). Prove that the ideal \((f(x))\) is not prime.

5. Test each polynomial below for reducibility over \(\mathbb{Q}\). If the polynomial is reducible, factor it completely. Otherwise, give a brief justification of why it is irreducible.
(a) \(x^5 + 9x^4 -6x^3 +3x - 15\)
(b) \(x^3 + 7x^2 + 1\)
(c) \(x^4 + 5x^3 + 5\)
(d) \(x^3 + 3x^2 + 2\)

Please typeset in LaTeX your solution to Exercise 4. Include a statement of the problem. This exercise will be graded both on correctness and presentation.

Homework 7, due Wednesday, April 4th:

Notation:
Let \(\mathbb{Z}/n\mathbb{Z}[x]\) denote the ring of all polynomials with coefficients in \(\mathbb{Z}/n\mathbb{Z}\).

If \(a\in R\), the ideal generated by \(a\) is denoted \((a)\) (in contrast to the notation in your textbook: \(\langle a \rangle\)).

Do, but do not hand in:
1. List all polynomials of degree 2 in the ring \(\mathbb{Z}/2\mathbb{Z}[x]\).

2. Show that the polynomial \(2x+1\) has a multiplicative inverse in the ring \(\mathbb{Z}/4\mathbb{Z}[x]\).

3. Show that the polynomial \(x^2 + 3x + 2\) has four zeros in the ring \(\mathbb{Z}/6\mathbb{Z}[x]\).

Hand in:
1. Let \(a\) be an element of a ring \(R\). Define the right annihilator of \(a\) as the set \[\textrm{Ann}(a) = \{x \in R \ : \ ax = 0\}.\] Prove that \(\textrm{Ann}(a)\) is a right ideal of R.

(You may refer to the previous homework set to assert that it's a subring.)

2. Prove that if \(A\) and \(B\) are ideals of a ring \(R\), then the set \[A + B = \{a + b \ : \ a \in A, b \in B\}\] is an ideal of \(R\).

3. Show that the quotient ring \(\mathbb{Z}/2\mathbb{Z}[x]/(x^2 + x + 1)\) is a field.

[Hint: Observe that \(x^2 = x+1\) in the quotient ring.]

4. Let \(R = \mathbb{Z}[x]\) and let \(I = (x^3 - 8)\).

(a) Find a polynomial of degree at most 2 that is congruent to \[x^7 - x^6 + 5x^5 - 3x^4 + 2x^3 + x^2 - 6x + 2\] in the quotient ring \(R/I\).

(b) Prove that \((x-2) + I\) is a zero divisor in \(R/I\).

Please typeset in LaTeX your solution to exercise 2. Include a statement of the problem. This exercise will be graded both on correctness and presentation.

Homework 6, due Wednesday, March 28th:

Do, but do not hand in:
1. Find a counterexample in a ring \(\mathbb{Z}/n\mathbb{Z}\) to each of the following common principles of the algebra of real numbers:

a) If \(ab = 0\), then \(a = 0\) or \(b = 0\).

b) If \(a^2 = a\), then \(a = 0\) or \(a = 1\).

c) If \(ab = ac\), then \(b = c\).

2. Prove that \(\mathbb{Z}[\sqrt{5}] = \{a + b\sqrt{5} \ : \ a,b \in \mathbb{Z}\}\) is an integral domain.

(Hint: To prove that it's a ring, observe that \(\mathbb{Z}[\sqrt{5}] \subseteq \mathbb{R}\) and use the subring test.)

Hand in:
1. Let \(a\) be an element of a ring \(R\). Define the right annihilator of \(a\) as the set \[\textrm{Ann}(a) = \{x \in R \ : \ ax = 0\}.\] Prove that \(\textrm{Ann}(a)\) is a subring of R.

2. Let \(R\) be a ring and let \(a \in R\). Prove that if \(a\) is a unit, then \(a\) is not a zero divisor.

3. Recall that \[\mathbb{Z} \oplus \mathbb{Q} = \{(z,q) \ : \ z \in \mathbb{Z} \textrm{ and } q \in \mathbb{Q}\}.\] Describe the set \(Z\) of all zero divisors and the set \(U\) of all units in the ring \(\mathbb{Z} \oplus \mathbb{Q}\).

4. Find a zero divisor in the ring \((\mathbb{Z}/5\mathbb{Z})[i] = \{a + bi \ : \ a,b \in \mathbb{Z}/5\mathbb{Z}\}\)
(where \(i\) is the usual imaginary unit, \(i = \sqrt{-1}\)).

5. Prove that \(\mathbb{Q}[\sqrt{5}] = \{a + b\sqrt{5} \ : \ a,b \in \mathbb{Q}\}\) is a field.

Please typeset in LaTeX your solution to exercise 1. Include a statement of the problem. This exercise will be graded both on correctness and presentation.

Homework 5, due Wednesday, March 7th:

Do, but do not hand in:
1. Prove that if \(H\) and \(K\) are both normal subgroups of \(G\), then \(H \cap K\) is a normal subgroup of \(G\).

Hand in:
1. A group of order \(p^n\) for some prime \(p\) and positive integer \(n\) is called a p-group. We will not get to this result, but it can be shown that every \(p\)-group has a nontrivial center -- i.e. if \(G\) is a \(p\)-group, then \(Z(G) \neq 1\) (where "\(1\)" is the commonly used abbreviation for the identity subgroup, \(\{1\}\)).

Let \(G\) be a non-abelian group of order \(p^3\) for some prime \(p\). Prove that \(|Z(G)| = p\).

(You will need the result stated above.)

2. Recall that if \(G_1\) and \(G_2\) are groups, the trivial homomorphism from \(G_1\) to \(G_2\) is the mapping that sends every element of \(G_1\) to the identity element of \(G_2\). Any other homomorphism is called nontrivial.

Suppose \(\phi : Z_{30} \to Z_5\) is a nontrivial homomorphim. Find \(\ker \phi\).

3. Let \(H\) and \(K\) be subgroups of \(G\), and define \[HK = \{hk \ | \ h \in H \text{ and } k \in K\}.\]
a) Let \(G \cong S_3\), \(H = \langle(1 \ 2)\rangle\), and \(K = \langle(2 \ 3)\rangle\). Find \(HK\) and explain why \(HK\) is not a subgroup of \(G\).

b) Prove that if \(H \trianglelefteq G\), then \(HK\) is a subgroup of \(G\).

c) If we replace the condition "\(H \trianglelefteq G\)" with "\(K \leq N_G(H)\)," how must your argument in (b) change? State the theorem that results.

4. Let \(H\) and \(K\) be normal subgroups of \(G\), and suppose \(H \cap K = 1\) (the identity subgroup) and \(G = HK\) (where \(HK\) is as defined in the previous exercise).

a) Prove that the mapping \(\phi(hk) = k\) (where \(h \in H\) and \(k \in K\)) is a homomorphism from \(G\) to \(G\).

b) Use the First Isomorphism Theorem to prove that \(G/H \cong K\).

[Hint: Use part (a).]

Please typeset in LaTeX your solution to exercise 4. Include a statement of the problem. This exercise will be graded both on correctness and on presentation.

Homework 4, due Friday, February 23rd:

Do, but do not hand in:
1. Find all of the left cosets of the subgroup \(H = \langle(1 \ 2 \ 3)\rangle\) of \(A_4\).

2. How many left cosets of the subgroup \(H = \langle(1 \ 2 \ 3 \ 4)\rangle\) are there in \(S_6\)?

3. Suppose a group \(G\) contains elements of every order from 1 through 10. What is the minimum possible order of \(G\)?

Hand in:

Notation: For any subset \(S = \{x_1, x_2, \ldots, x_n\}\) of any group \(G\), we call the smallest subgroup containing \(S\) the subgroup generated by \(S\), denoted \(\langle x_1, x_2, \ldots, x_n\rangle\) or \(\langle S\rangle\). (This is an extension of the familiar notation for cyclic groups and subgroups, which are generated by a single element.)

1. Let \(p\) and \(q\) be distinct primes, and let \(G\) be a group of order \(pq\). Prove that if \(x\) and \(y\) are nonidentity elements of \(G\) and \(|x| \neq |y|\), then \(\langle x,y \rangle = G\).

2. Let \(p\) be a prime and let \(G\) be a group of order \(p^2\). Prove that either \(G\) is cyclic or \(g^p = 1\) for every \(g \in G\).

3. Consider the permutations \(\sigma = (1 \ 2)\) and \(\tau = (2 \ 3 \ 4)\) as elements of \(S_4\). Prove that \(\langle \sigma, \tau \rangle = S_4\).

4. Let \(H \leq G\). Prove that \(C_G(H) \trianglelefteq N_G(H)\).

(We proved in class that \(C_G(H) \leq N_G(H)\), so you need only prove that \(C_G(H)\) is normal in \(N_G(H)\)).

Please typeset in LaTeX your solution to exercise 4. Include a statement of the problem. (This exercise will be graded both on correctness and on presentation.)

Homework 3, due Wednesday, February 14th:

Do, but do not hand in:
1. Prove that \(\left|x^{-1}\right| = \left|x\right|\) for any element \(x\) of any group \(G\).

2. Let \(A = \begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}\) and \(B = \begin{bmatrix}0 & 1\\-1 & -1\end{bmatrix}\) in \(\textrm{SL}_2(\mathbb{R})\). Find \(|A|\), \(|B|\), and \(|AB|\).

3. Let \(G = \langle a \rangle\) and let \(|a| = 24\).
a) List all generators for the subgroup of order 8.
b) List all generators for \(G\).

4. Write each of the following as a product of disjoint cycles:
a) (1 2 3 4)(1 3 5)
b) (1 2 3 4)(1 2 3)
c) (1 5)(1 4)(1 3)(1 2)
d) (1 2 3 4)(1 3)
e) (1 2 3 4)(1 4 3)
f) (1 2 3 4)(3 4 5 6)

5. Find the order of each of the permutations in exercise 4.

6. Determine whether each of the permutations in exercise 4 is even or odd.

7. Prove that \(A_8\) contains an element of order 15.

Hand in:
1. Give the subgroup lattice for \(Z_{12} = \langle a \rangle\).

2. Let \(G = \langle a \rangle\) with \(|G| = n\). Prove that \(g^n = 1\) for every \(g \in G\).

3. Prove that \(S_n\) is non-abelian for all \(n \geq 3\), and that \(A_n\) is non-abelian for all \(n \geq 4\).

4. Let \(G\) be a group of permutations on a set \(\Omega\). Let \(a \in \Omega\), and define the stabilizer in \(G\) of \(a\) as: \[G_a = \{\sigma \in G \ : \ \sigma(a) = a\}.\] Prove that \(G_a\) is a subgroup of \(G\).

Note: Gallian calls this subgroup "stab(\(a\))." His solution, which is probably in your textbook (and which you are welcome to look at), is both terse and incomplete. Make sure to give a complete solution in your own words.

Please typeset in LaTeX your solution to exercise 4. (This exercise will be graded for both correctness and presentation.) You may use this LaTeX file as a template, if you like.

Homework 2, due Monday, February 5th:

1. Let \(H\) and \(K\) be subgroups of \(G\). Prove that \(H \cap K\) is a subgroup of \(G\).

2. Let \(H\) and \(K\) be subgroups of \(G\). Prove that \(H \cup K\) is a subgroup of \(G\) if and only if either \(H \subseteq K\) or \(K \subseteq H\).

3. Let \(a\) be an element of a group \(G\). Prove that \(C_G(a) = C_G(a^{-1})\).

4. Prove that if \(H \leq G\), \(K \leq G\), and \(H \subseteq K\), then \(H \leq K\).

5. Let \(H\) be a subgroup of \(G\). Prove that \(H \leq C_G(H)\) if and only if \(H\) is abelian.

6. Let \(G\) be a group.
(a) Let \(H \leq G\). Show that \(C_G(H) = \{g \in g \ \mid \ g^{-1}hg = h \textrm{ for all } h \in H\}\).

(b) Let \(g, h \in G\). Prove that \((g^{-1}hg)^n = g^{-1}h^n g\) for any \(n \in \mathbb{N}\).

(c) Conclude that \(C_G(h) \leq C_G(h^n)\) for any \(n \in \mathbb{N}\).
(In other words, if \(g\) commutes with \(h\), then \(g\) commutes with all powers of \(h\).)

Homework 1, due Friday, 1/26:

1. Prove that the operation \(*\) defined on \(\mathbb{Q} - \{0\}\) by \(a * b = \displaystyle \frac{a}{b}\) is not associative.

2. Prove that the operation \(*\) defined on \(\mathbb{Q}\) by \(a * b = a + b + ab\) is associative.

3. Suppose \(x,y,z \in G\) where \(G\) is a group (under the operation of multiplication). What is the inverse of the element \(xyz\)?

4. Suppose \(G\) is a group (under the operation of multiplication) with identity \(1\) such that \(x^2 = 1\) for every \(x \in G\). Prove that \(G\) is abelian.

5. Let \(G = \{a + b \sqrt{2} \mid a,b \in \mathbb{Q}\}\).
(a) Prove that \(G\) is a group under addition.
(b) Prove that \(G - \{0\}\) is a group under multiplication.

[For both parts, show that the operation is in fact a binary operation on \(G\) -- i.e. \(x, y \in G\) implies \(x*y \in G\). Also (for both parts), you can argue that associativity is inherited from \(\mathbb{R}\) (since \(G \subset \mathbb{R}\), \(x, y, z \in G\) implies \(x, y, z \in \mathbb{R}\)).]