primitive root of unity

**apple2009** · April 24th 2010, 09:31 PM

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.

**aliceinwonderland** · April 25th 2010, 01:04 AM

Originally Posted by apple2009

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.

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