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Math Help - primitive root of unity

  1. #1
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    primitive root of unity

    Let x=e^(2∏/13i), a primitive 13th root of unity.
    Find a subfield K of Q(X) with (Q(x):K)=3.
    Find a subfield L of Q(X) with (Q(x):L)=4.
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  2. #2
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    Quote Originally Posted by apple2009 View Post
    Let x=e^(2∏/13i), a primitive 13th root of unity.
    Find a subfield K of Q(X) with (Q(x):K)=3.
    Find a subfield L of Q(X) with (Q(x):L)=4.
    Let \zeta be the primitive 13th roots of unity. We see that \text{Gal}(\mathbb{Q}(\zeta), \mathbb{Q})=(\mathbb{Z}/13\mathbb{Z})^\times \cong \mathbb{Z}/12\mathbb{Z}, where the generator of this cyclic group is \sigma:\zeta \mapsto \zeta^2.

    You need to find the subgroup of the order 3 and order 4 for the above group and correspond them to the subfields of \mathbb{Q}(\zeta).

    The subgroup of order 3 for \text{Gal}(\mathbb{Q}(\zeta), \mathbb{Q}) is generated by \sigma^4.

    Thus K=\mathbb{Q}(\zeta + \sigma^4(\zeta) + \sigma^8(\zeta)) = \mathbb{Q}(\zeta + \zeta^{2^4} + \zeta^{2^8}) = \mathbb{Q}(\zeta + \zeta^3 + \zeta^9), where [\mathbb{Q}(\zeta):K ] = 3.

    You can find the L exactly in the same way.
    Last edited by aliceinwonderland; April 25th 2010 at 01:24 PM.
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