- Problem Set 12, Monday 05/09/2016 (complete)
- Prove that any extension of degree 2 is a normal
extension.
- Let E/F be a Galois extension, and L an
intermediate subfield, such that L/F is a normal
extension. Prove that the restriction map
ρ:AutF(E) → AutF(L)
given by ρ(τ) = τ|L, is a
surjective homomorphism.
- Consider the polynomial f=x5 - 20 x + 6
∈ ℚ[x].
- Show that f is irreducible
over ℚ.
- Show that f has exactly three real
roots. (Hint: use the Intermediate Value Theorem from
Calculus).
- If E is the splitting field of f
over ℚ, show that [E:ℚ]
is divisible by 5.
Some additional problems. Will not to be collected.
- Let p be a prime number. In the permutation
group Sp, let ρ be a p-cycle and
let σ be a transposition. Show
that Sp is generated by {ρ, σ}.
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- Problem Set 11, Monday 05/02/2016 (complete)
- Let E be a field, G a subgroup
of Aut(E), F=EG and L ∈
SubF(E). Show that L* =
AutL(E) , and it is a subgroup of G.
- Prove the lemma stated in class about the Galois
connection, i.e. that (*) is order-reversing and that (1 ≤ **).
Some additional problems. Will not to be collected.
- Let F = ℤ2(t) be the field of
rational functions over ℤ2 on the
variable t. Let E be the splitting field
of x2 - t ∈ F[x]. Is E/F a
Galois extension? Explain why.
- let α be the only real root of
x5-2 ∈ ℚ[x]. Explain why
ℚ(α)/ℚ is not a Galois
extension. Find an extension E
of ℚ(α) such that
E/ℚ is a Galois extension. Can you
find E minimal with this property?
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- Friday 05/06/2016 (complete)
- Let E be a field, G a subgroup
of Aut(E), F=EG and L ∈
SubF(E). Show that L* =
AutL(E) , and it is a subgroup of G.
- Prove the lemma stated in class about the Galois
connection, i.e. that (*) is order-reversing and that (1 ≤ **).
- Let F = ℤ2(t) be the field of
rational functions over ℤ2 on the
variable t. Let E be the splitting field
of x2 - t ∈ F[x]. Is E/F a
Galois extension? Explain why.
- let α be the only real root of
x5-2 ∈ ℚ[x]. Explain why
ℚ(α)/ℚ is not a Galois
extension. Find an extension E
of ℚ(α) such that
E/ℚ is a Galois extension. Can you
find E minimal with this property?
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