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Topic: about the Kronecker-Weber theorem
Replies: 6   Last Post: Feb 4, 2013 7:45 PM

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Posts: 11,254
Registered: 7/15/05
Re: about the Kronecker-Weber theorem
Posted: Feb 4, 2013 4:04 AM
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David Bernier wrote:
>I have a further question about conjugate roots ...
>The non-trivial third roots of unity
>-1/2 +i*srqrt(3)/2 and -1/2 -i*srqrt(3)/2
>are complex conjugates.
>I don't know of a definition where, for example, in the
>setting above,
>2^(1/3) is said to be conjugate to
>2^(1/3) * (-1/2 +i*srqrt(3)/2).

Let H be an algebraically closed field, and let K be a
subfield of H. Let K[x] denote the ring of all polynomials
in the indeterminate x with coefficients in K. If f in K[x]
is irreducible in K[x], the roots of f in H are said to be
conjugates of each other over K.

Thus, the 3 cube roots of 2 are conjugate over Q since
they are roots of the polynomial x^3 - 2 which is irreducible
over Q.


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