Are the following true or false?1) ∀n∈N:[Q(exp⁡(2πin):Q]=n−12) Let f∈Q[X] be irreducible, deg(f)=n. Then: |Gal(f)|=n3) Let K(a}K be an algebraic field extension. Then Gal(K(a)K) is commutative4) Let LK be a finite field extension and |Gal(LK)|=1. Then LK is normal.

Step 11) Try with n=42) Try with f=X3−2∈Q[X]3 and 4) Set K=Q, and take L=Q(3{2})

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Juliet Jackson

2022-03-17

Are the following true or false?
1)

\forall n \in N : [Q (\exp (2 π \frac{i}{n}) : Q] = n - 1

2) Let

f \in Q [X]

be irreducible,

d e g (f) = n

. Then:

| G a l (f) | = n

3) Let

\frac{K (a}}{K}

be an algebraic field extension. Then

G a l (\frac{K (a)}{K})

is commutative
4) Let

\frac{L}{K}

be a finite field extension and

| G a l (\frac{L}{K}) | = 1

. Then

\frac{L}{K}

is normal.

Tavolillaqra

Beginner2022-03-18Added 4 answers

Step 1
1) Try with

n = 4

2) Try with

f = X^{3} - 2 \in Q [X]

3 and 4) Set

K = Q

, and take

L = Q (\sqrt{3} {2})

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