Two intermediate field E and L are said to be conjugate if there exists auch that . Prove that E and L are conjugate if and only if and are conjugate subgroup of role="math" localid="1657965474587" .
and are conjugate subgroup of .
An abelian group is a commutative group and satisfy the condition for all in the field extension is .
Consider the fixed subgroups of the extension field and this normal fixed subgroup consist two subgroup and here the conditions is given both subgroup over the fixed are conjugate.
Now the property of conjugate between two subgroup if it is exist so the Galois group of the fixed field over the extension field is conjugate. Consider the two element of the subgroup over the if the subgroup are conjugate so all the element to the subgroup are conjugate so both element are conjugate and the automorphism on is define as and minimal polynomial of over is role="math" localid="1657965967250" assume both element are zeros of the same irreducible polynomial over the fixed field .
Then is isomorphism of the element is
And automorphism of the element is
Same for another subgroup consist the two element and automorphism is
Since the element of the both subgroup are conjugate and subgroup are also conjugate so the automorphism over the fixed field is
If both subgroup over the fixed field are conjugate so Galois field of the both subgroup over the extension field are also conjugate by using the galois theory.
Since and are conjugate if and only if and are conjugate of the field over the extension field is .
94% of StudySmarter users get better grades.Sign up for free