Website owner: James Miller
Prove. An automorphism of the Galois group carries a root of the given polynomial equation into another root.
Proof. Let P be the ground field and K be the splitting field. A automorphism is defined as a one-to-one mapping A of the splitting field K onto itself with the properties
A1) (a + b)A = aA + bA
A2) (ab)A = aA⋅bA
A3) αA = α
where a,b ∈ K, α ∈ P
Let x = α be a root of the equation
1) xn + a1xn-1 + .... + an = 0
Then
2) αn + a1αn-1 + .... + an = 0
Let us operate on both sides of 2) with the automorphism A and apply the properties A1 - A3
3) (αn + a1αn-1 + .... + an )A = 0 ⋅A
Using property A1, we get
4) αnA + a1αn-1A + .... + an A = 0 ⋅A
Using properties A2 and A3 we get
5) (αA)n + a1(αA)n-1 + .... + an A = 0 ⋅A
Since 0 ⋅A = 0 and aiA = ai we get
6) (αA)n + a1(αA)n-1 + .... + an = 0
This last equation proves that an automorphism carries one root into another root. If we let β = αA, 6) can be written
7) (β)n + a1(β)n-1 + .... + an = 0
and we have shown that an automorphism takes one root α into another root β.
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