Math 430 - Homework

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Homework

Suggested practice on Galois theory

A. For each of the splitting fields K in p227 #6, determine Gal(K/Q) and the lattice of intermediate fields.

B. Let K=Q(√p₁,...,√p_k). Show that Gal(K/Q) is isomorphic to (Z₂)^k.

C. If F contained in the constructible numbers, and F is a finite extension over Q, what can you say about the group structure of Gal(F/Q)?

Notation in all: let Q denote the rational numbers, √p be the square root of p, and Z₂ be the integers mod 2.

Homework 12, due Apr 12 at 12:07pm

Please do the following problems from the textbook:

Herstein, p227: 5, 6, 7, 10, 14
Herstein, p231: 1, 2, 3, 4, 5 (note that F is any subfield of the reals)

Hint on p231 #1: similar triangles may be helpful.

Homework 11, due Apr 15 at 12:07pm

Please do the following problems from the textbook:

Herstein, p214: 5, 8, 10
Herstein, p219: 2
Herstein, p227: 11, 12, 13

and the following additional problems:

A. If K is an algebraic field extension of F, give the details of the induction proof that F(a₁, a₂, ..., a_n) is a finite degree extension of F (where each a_i is in K).

B. Let f and g be polynomials in F[x], and that f divides g in some K[x] for some extension field K of F. Show that f divides g in F[x].

C. Let f be an irreducible polynomial in F[x] of degree p. Suppose K is a algebraic field extension of F such that f is not irreducible in K[x]. Show that p divides [K:F].
Hint: Consider an extension field L of K such that f has a root in L[x].

D. (More challenging) Show that if a and b are elements of a field extension K over F, then a is algebraic over F(b) if and only if b is algebraic over F(a).
Hint: Look at the extension field F(a,b) over F(a).

Homework 10, due Apr 8 at 12:07pm

Please do the following problems from the textbook:

Herstein, p167: 1, 2
Herstein, p214: 3, 4, 9

and the following additional problems:

A. Show that the face ring of any simplicial complex (over F) is an F-algebra.

Homework 10, due Apr 3 (Friday) at 12:07pm

Please do the following problems from the textbook:

Herstein, p161: 3, 4, 5
Herstein, p167: 6, 16, 17

Homework 9, due Mar 25 at 12:07pm

Please do the following problems from the textbook:

Herstein, p142: 5, 6
Herstein, p149: 3, 7, 8
Herstein, p152: 1, 2, 3, 8
Herstein, p158: 1, 2, 3, 7
Herstein, p166: 8

Homework 8, due Mar 18 at 12:07pm

Please do the following problems from the textbook:

Herstein, p130: 4, 6, 7, 8
Herstein, p135: 5, 6, 7, 9, 10, 12
Herstein, p142: 2, 3, 4

and the following additional problems:

A. Explain why the ring in p135 #12 is not a field or division ring.

B. ~~Exercise 1~~Exercise 6 from the Zorn's Lemma and Maximal Ideals handout.

Homework 7, due Mar 4 at 12:07pm

Please do the following problems from the textbook:

Herstein, p102: 17
Herstein, p108: 4, 5, 6, 14, 16
Herstein, p116: 11, 13
Herstein, p130: 9, 10

(Hint: On p102 #17, remember that normal Sylow subgroups are characteristic.)

and the following additional problems:

A. Show that there are no nonabelian simple groups of order < 101, with the exception of the simple group of order 60, and possibly orders 90 and 96.

B. (More challenging) Show that there are no simple groups of order 90 or 96.

Homework 6, due Feb 25 at 12:07pm

Please do the following problems from the textbook:

Herstein, p80: 9, 10, 11, 21a-c
Herstein, p90: 3, 9, 10
Herstein, p102: 7, 12a-b, 13

(Hint: On p90 #9, the Sylow C-Theorem may save you some work!)

and the following additional problems:

A. Explain why p-groups cannot be simple and nonabelian.

B. Explain why no groups of order < 24 are nonabelian simple.

Homework 5, due Feb _20_ at 12:07pm

Please do the following problems from the textbook:

Herstein, p80: 2, 3, 4, 5, 8a
Herstein, p90: 2
Actions handout: Exercise 21, Exercise 37

and the following additional problems:

A. Using your result from p90 #2, find a normal subgroup of order 4 in S₄.

B. What is the kernel of the conjugation action of G on G?

C. Show that S₆ has a subgroup of index 6. (Hint: use an action!)

D. Let G be a simple group of order 360. Show that G has no subgroups of index 2, 3, 4, or 5. (You might be interested to know that G is a subgroup of S₆, and has a subgroup of index 6 for a reason similar to C.)

Homework 4, due Feb 11 at 12:07pm

Please do the following problems from the textbook:

Herstein, p65: 15
Herstein, p70: 1, 4, 5, 7, 8, 9, 21
Herstein, p75: 6, 7
Herstein, p90: 8 (Use Lemma 13 from the handout)

and the following additional problems:

A. Let G be a finite group, and T be a homomorphism from G to K, and let H be a subgroup of G. Show that [T(G) : T(H)] divides [G : H ], and |T(H)| divides |H|.

B. For what groups G is the map I defined by I(g)=g^-1 an automorphism?

C. p70, problem 14.
Hint: there are two classes of groups that you may want to handle as different cases.

Homework 3, due Feb 4 at 12:07pm

Please do the following problems from the textbook:

Herstein, p53: 4, 5, 12, 13, 16
Herstein, p65: 1, 5, 11, 16, 18, 19

and the following additional problems:

A. Let G be finite and abelian, and let m divide |G|. Show that G has a subgroup of order m.
Hint: Apply Cauchy's Theorem for Abelian Groups inductively.

Homework 2, due Jan 28 at 12:07pm

Please do the following problems from the textbook:

Herstein, p46: 2, 3, 6, 8, 10, 11, 14
Herstein, p53: 1, 2, 10, 11

(Hint on p46 #2: Assume G has an element of infinite order...)

and the following additional problems:

A. Let D₁₀ be the dihedral group with 10 elements, that is, all rotations and flips of the pentagon. Describe all subgroups of D₁₀. Which subgroups are normal?

B. Let H, K, and L be subgroups of G, with H⊆K. Prove that K∩HL=H(K∩L).

Homework 1, due Jan 21 at 12:07pm

Please do the following problems from the textbook:
Herstein, p35: 1, 3, 7, 9, 10, 12, 13, 24
Herstein, p46: 1, 4
and the following additional problems:
A. In Herstein p35 #24, the resulting group G is of order 6. S₃ is also of order 6. Compare and contrast G with S₃.
B. Suppose that G=H∪K where H and K are subgroups of G. Show that either H=G or K=G. (Red text added later.)