Notice that the conjugate fields of are always isomorphic but not necessarily distinct.
All conjugate fields are equal, i.e. (http://planetmath.org/Ie) , or equivalently belong to , if and only if the extension is a Galois extension of fields. The reason for this is that if is an algebraic number and is the minimal polynomial of then the roots of are precisely the algebraic conjugates of .
For example, let . Then its only conjugate is and is Galois and contains both and . Similarly, let be a prime and let be a primitive th root of unity (http://planetmath.org/PrimitiveRootOfUnity). Then the algebraic conjugates of are and so all conjugate fields are equal to and the extension is Galois. It is a cyclotomic extension of .
Now let and let be a primitive rd root of unity (i.e. is a root of , so we can pick ). Then the conjugates of are , , and . The three conjugate fields , , and are distinct in this case. The Galois closure of each of these fields is .
|Date of creation||2013-03-22 17:10:28|
|Last modified on||2013-03-22 17:10:28|
|Last modified by||pahio (2872)|