please write the answer and logic clearly.

here is the hw and some useful handout

MAT 150C: MODERN ALGEBRA

Homework 9

Instructions. Please write the answer to each problem, including the computational ones, in connected

sentences and explain your work. Just the answer (correct or not) is not enough. Write your name in every

page and upload to Gradescope with the correct orientation. Make sure to indicate to Gradescope which

pages correspond to each problem. Finally, if you used another sources or discussed the problem

with classmates, be sure to acknowledge it in your homework.

1. Let F ⊆ K be a Galois extension. For α ∈ K, define the norm of α over F to be:

Y

NK/F (α) :=

ϕ(α) ∈ K

ϕ∈G(K/F)

and the trace of α:

X

TrK/F (α) :=

ϕ(α) ∈ K.

ϕ∈G(K/F)

(a) Show that NK/F (α) and TrK/F (α) are in F.

(b) Consider F = Q and K = Q(ζp ), where p is prime and ζp = e2π

TrK/F (ζp ).

√

−1/p .

Find NK/F (ζp ) and

(c) More generally, assume that K = F(α) is a Galois extension, and that

irrF (α) = xn + a1 xn−1 + · · · + a1 x + a0 ∈ F[x]

show that NK/F (α) = (−1)n a0 and that TrK/F (α) = −a1 .

(d) OPTIONAL: In the setting of part (c), let Mα : K → K be the map Mα (β) = αβ. Note that

this map is F-linear, so we can talk about its trace, determinant, characteristic polynomial etc.

Show that NK/F (α) is the determinant of Mα ; and TrK/F (α) is the trace of Mα . (Hint: Use the

Cayley-Hamilton theorem to compare the polynomial irrF (α) to the characteristic polynomial of

Mα .)

2. List all the intermediate fields of the field extension Q ⊆ Q(ζ7 ), where ζ7 = e2π

√

−1/7 .

3. Let n > 0, and consider the symmetric group Sn .

(a) Show that for 0 ≤ k ≤ n − 2, (12 · · · n)k (12)(12 · · · n)−k = (k + 1, k + 2).

(b) Show that if H ⊆ Sn is a subgroup and (12), (12 · · · n) ∈ H, then H = Sn .

(c) Let H ⊆ S5 be a subgroup. Show that if H contains a transposition and a 5-cycle, then H = S5 .

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