MAT 510 University of Nairobi Mathematics Homework 9 please write the answer and logic clearly. here is the hw and some useful handout MAT 150C: MODERN ALG

MAT 510 University of Nairobi Mathematics Homework 9 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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