MATH 322 Homeworks

First homework

Reading assignment:

§38 Symmetric polynomials

Written Assignment:

(1) Express the symmetric polynomial

over Z in terms of the elementary symmetric polynomials.

(2) Find a polynomial over Z whose roots are the squares of the roots of t³ + 5t² + 7t + 1 in Z[t].

Second Homework

Reading assignment:

§39 Vector spaces, §40 Subpaces, §41 Factor spaces.

Written Assignment (due Feb. 27, 2001):

(3) Determine whether Q x Q is a vector space if the operations are defined by

for all a, b, c, d in Q.

(4) If q is a positive integer and K is a field with q elements, how many elements does Kⁿ have?

Third Homework

Reading assignment:

§41 Factor spaces, §42 Dependence and Bases

Written Assignment (due March 13, 2001):

(5) Find all Z₂-bases of Z₂³.

Fourth Homework

Reading assignment:

§48 Field Extensions, §49 Field Extensions (continued)

Written Assignment (due March 20, 2001):

(6) Let K be a field and let Aut(K) be the set of all field automorphisms of K. Show that Aut(K) is a group under composition.

(7) Find all automorphisms of Q, Z_p, Q(w).

(8) Find the degrees of the following extensions: C/R, C/Q(i), R/Q, F_p(x)/F_p.

Fifth Homework

Reading assignment:

§50 Algebraic Extensions, §51 Kronecker's Theorem

Written Assignment (due March 27, 2001):

(9) (§50 Ex. 1 (e)) Find the minimal polynomial of ³√2 + √2 over Q(³√2), over Q(√2), over Q.

Sixth Homework

Reading assignment:

§52 Finite Fields

Written Assignment (due April 3, 2001):

(10) Find a field of 125 elements. Describe its elements, explain how they are added and multiplied.

Seventh Homework

Reading assignment:

§52 Finite Fields, §53 Splitting Fields (up to and including 53.8)

Written Assignment (due April 12, 2001):

(11) Prove that a root of x² + 7x + 2 in F₁₁[x] is a generator of F₁₂₁^×.

Eighth Homework

Reading assignment:

§54 Galois Theory

Written Assignment (due April 19, 2001):

(12) Let E/K be a field extension, and write G := Aut_KE for short. Prove that, if H is a normal subgroup of G, then H'' is also a normal subgroup of G.

Ninth Homework

Reading assignment:

§54 Galois Theory

Written Assignment (due April 26, 2001):

(13) Let E/K be an arbitrary field extension and let G = Aut_KE. Prove that E is a Galois extension of G'.

Tenth Homework

Reading assignment:

§55.1, 55.2, 55.3, 55.4, 55.5, 55.6, 55.7, 55.8, 55.9, 55.10, 55.13, 55.14, 55.20, 55.21

Written Assignment (due May 15, 2001):

(14) Find a splitting field K over F₃ of  (x² + 1)(x² + x + 2) and a primitive element of K.

Eleventh Homework

Reading assignment:

§58

Written Assignment (due May 24, 2001):

(15) (§58 Ex. 4) Let p, k be natural numbers, p prime. Let F_p(x) denote the p^th cyclotomic polynomial over Q. Prove that, if d divides F_p(k), then d = p or d is congruent to 1 mod p.

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