Let-f-x-Q-x-irreducible-of-degree-n-and-K-it-s-Splitting-Field-over-Q-Prove-that-if-Gal-K-Q-is-Abeilan-then-Gal-K-Q-n-How-can-i-prove-this-

Question Number 213208 by issac last updated on 01/Nov/24

Let f (x) \in Q [x] irreducible of degree n

and K {it}^{'} s Splitting Field over Q

Prove that if Gal (K ∖ Q) is Abeilan

then ∣ Gal (K ∖ Q) ∣= n

How can i prove this ? ? ?

Answered by MrGaster last updated on 01/Nov/24

f (x) \in Q [x], \deg (f) = n

K / Q is the splitting field of f (x) = G = G al (K / Q)

G is abelian

α_{1}, α_{2}, \dots, α_{n} are roots of f (x) in K

K = Q (α_{1}, α_{2}, \dots, α_{n})

∣ G ∣= [K : Q]

σ \in G, σ (α_{i}) = α_{j}, σ (α_{j}) = α_{i}, σ (α_{κ}) = α_{κ}, k \neq i, j

σ τ = τ σ, \forall σ, τ \in G

σ (α_{i}) α_{j} \Rightarrow σ (α_{j}) = α_{i} or α_{j}

σ (α_{i}) = α_{j}, τ (α_{i}) = α_{κ}, σ (α_{κ}) = α_{κ}

σ τ (α_{i}) = σ (α_{κ}) = α_{κ}

τ σ (α_{i}) = τ (α_{j}) = α_{κ}

σ (α_{j}) = α_{j} for all σ \in G

σ (α_{i}) = α_{j} \Rightarrow α_{i} and α_{j} are conjugate over Q α_{1}, α_{2}, \dots, α_{n} are distinct and conjugate o

∣ G ∣= n

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