conjugate fields

If  ϑ1,ϑ2,,ϑn  are the algebraic conjugates of the algebraic numberMathworldPlanetmath ϑ1, then the algebraic number fieldsMathworldPlanetmath(ϑ1),(ϑ2),,(ϑn)  are the conjugate fields of (ϑ1).

Notice that the conjugate fields of (ϑ1) are always isomorphicPlanetmathPlanetmathPlanetmath but not necessarily distinct.

All conjugate fields are equal, i.e. ( (ϑ1)=(ϑ2)==(ϑn), or equivalently ϑ1,,ϑn belong to (ϑ1), if and only if the extensionPlanetmathPlanetmath (ϑ1)/ is a Galois extensionMathworldPlanetmath of fields. The reason for this is that if ϑ1 is an algebraic number and m(x) is the minimal polynomialPlanetmathPlanetmath of ϑ1 then the roots of m(x) are precisely the algebraic conjugates of ϑ1.

For example, let ϑ1=2. Then its only conjugatePlanetmathPlanetmathPlanetmath is ϑ2=-2 and (2) is Galois and contains both ϑ1 and ϑ2. Similarly, let p be a prime and let ϑ1=ζ be a primitive pth root of unityMathworldPlanetmath ( Then the algebraic conjugates of ζ are ζ2,,ζp-1 and so all conjugate fields are equal to (ζ) and the extension (ζ)/ is Galois. It is a cyclotomic extension of .

Now let ϑ1=23 and let ζ be a primitive 3rd root of unity (i.e. ζ is a root of x2+x+1, so we can pick ζ=-1+-32). Then the conjugates of ϑ1 are ϑ1, ϑ2=ζ23, and ϑ3=ζ223. The three conjugate fields (ϑ1), (ϑ2), and (ϑ3) are distinct in this case. The Galois closure of each of these fields is (ζ,23).

Title conjugate fields
Canonical name ConjugateFields
Date of creation 2013-03-22 17:10:28
Last modified on 2013-03-22 17:10:28
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 10
Author pahio (2872)
Entry type Definition
Classification msc 12F05
Classification msc 11R04
Related topic PropertiesOfMathbbQvarthetaConjugates