MATH:5010 - Spring 2017

Homework Assignment 11

Reading (from Dummit and Foote):

Read Sections 14.5 (Theorem 26), 14.3, 14.4 (we proved the Primitive Element Theorem differently), 14.7, 14.6 (p. 606-611).

Please look through the sections for EACH lecture before the lecture. See the Lecture Schedule for the sections that will be covered.

Then after EACH lecture, review your lecture notes and thoroughly read the sections covered in class.

Part A Problems (to be handed in on Tuesday, April 25, at the beginning of the discussion section):

Section 14.2: # 17. (You should be able to do both #17 and #18 but you only have to hand in #17.)
IMPORTANT:
- There are some typos in #17.
  You should assume that there exists a FINITE Galois extension L/F such that L contains K. You do not have to worry about the existence of L, but you just assume it exists.
  (We have shown in class on April 18 that if K/F is finite separable then L exists. Conversely, if L exists then L/F is separable, which implies that K/F is separable.)
  They do NOT say that L is a Galois closure of K over F, so do NOT assume that.
  Fix an algebraic closure of F. The product defining the norm should be over all embeddings σ of K into this fixed algebraic closure such that σ|_F = id_F.
- For part (a), show that every τ ∈ Gal(L/F) fixes N_K/F(α).
- In part (c), assume D ∈ F is not a square in F.
- Part (d) is independent of part (c) and could have been put before (c). In particular, K is as at the beginning of the problem, i.e., K is an arbitrary finite (separable) extension of F such that there exists a finite Galois extension L/F with L containing K.
Section 14.3: # 8.
HINT for #8: Recall 13.5 #5 which was on Homework 9.
Section 14.5: # 7, 8.
IMPORTANT for #7: ζ_n denotes a primitive n-th root of unity in C. σ_-1 is the notation introduced in Theorem 26 on p. 596 with a = -1. The subfield of real elements in K is the intersection of K and the real numbers R. To prove the last part of the problem, it may be helpful to show that K⁺ is the fixed field of σ_-1 and to find the minimal polynomial of ζ_n over Q(ζ_n+ζ_n^-1). You may freely use Theorem 26 on p. 596 (we proved this in class).
IMPORTANT for #8: To make the notation easier, please assume that for all n ≥ 0 we have ζ_2ⁿ⁺³² = ζ_2ⁿ⁺². In part (c), in the displayed equation for α_n you should replace all the plus signs by plus or minus and also put plus or minus in front of the overall square root symbol. (This is to indicate that square roots are only unique up to plus/minus.)

Part B Problems (to be handed in on Friday, April 28, BY PUTTING THEM INTO ERIK'S MAILBOX BY 12:30PM):

If you hand in ALL Part B Problems together with the Part A Problems already on Tuesday, April 25, 2 extra points will be added to your homework score.

Section 14.3: # 11.
IMPORTANT: You need to show that if x^pⁿ - x + 1 is irreducible over F_p, then n = 1 or n = p = 2. To use the hint given in the text, fix an algebraic closure of F_p containing both α and F_pⁿ and use that for each positive integer r there is a unique field inside this algebraic closure of order p^r. To show [F_p(α):F_pⁿ] = p, use that F_p(α)/F_p is Galois and that the Galois group is cyclic.
Section 14.4: # 4.
IMPORTANT: Assume that f(x) is separable and irreducible in F[x] (otherwise the splitting field L of f(x) over F may not be Galois as is assumed later in the hint given in the text). Assume that K/F is a finite Galois extension and that K and L lie inside a common field (this can be easily arranged by taking an algebraic closure of K and then taking the splitting field L of f(x) over F inside this algebraic closure). You only need to use Section 4.1, #9(a) (note that Gal(L/F) is finite). To show the formula for m, Corollary 20 on p. 592 may be helpful.
Section 14.7: # 4.
IMPORTANT: Assume first that the n^th root of a you adjoin to Q to obtain K is a positive real number. Hence, also the d^th root of a showing up should be a positive real number since E is supposed to be a subfield of K. After you have done this case, do the general case, i.e. let α be an arbitrary root of xⁿ - a and show that if E is a subfield of K = Q(α) with [E:Q] = d then E = Q(α^n/d).

Additional Practice Problems (you have to know how to do these, but you do NOT have to hand these in):

Section 14.2: # 3, 6, 8, 10, 11, 18, 19, 20, 31.
Section 14.3: # 3, 6.
Section 14.5: # 4, 5, 6.
Section 14.6: # 4, 5, 11, 17.

Frauke Bleher
Apr 18 2017