[Math] Number of subfields of splitting fields of $x^5-5$ over $\mathbb{Q}$.

abstract-algebragalois-theory

I want to find the number of subfields of splitting fields of $x^5-5$ over $\mathbb{Q}$.

  1. By Eisenstein Criterion, $x^5-5$ is irreducible over $\mathbb{Q}.$

  2. Then splitting field $K$ of $x^5-5$ is Galois extension of $\mathbb{Q}$.

  3. Let $\zeta$ be primitive 5th root. Then Roots of $x^5-5$ consist of exactly $\zeta \root5\of5$, $\zeta^2 \root5\of5$, $\zeta^3 \root5\of5$, $\zeta^4 \root5\of5$, $\root5\of5$.
    So $$K=\mathbb{Q}(\zeta \root5\of5, \zeta^2 \root5\of5, \zeta^3 \root5\of5, \zeta^4 \root5\of5) = \mathbb{Q}(\zeta,\root5\of5)$$

  4. Since $K=\mathbb{Q}(\zeta)\mathbb{Q}(\root5\of5)$ and $[\mathbb{Q}(\zeta):\mathbb{Q}] = 4 \mbox{ and } [\mathbb{Q}(\root5\of5):\mathbb{Q}]=5$,

    $$|Gal(K/F)|=[K:F]=5\cdot 4=20.$$

Therefore $Gal(K/F)$ is group of order 20.

So If I can find number of subgroups of $Gal(K/F)$, then I can find number of subfields of the field.

What shall I do?

Best Answer

Since $K=\mathbb{Q}(\zeta,\sqrt[5]{5})$, an element $\sigma\in\mathrm{Gal}(K/\mathbb{Q})$ is completely determined by what $\sigma(\zeta)$ and $\sigma(\sqrt[5]{5})$ are. Note that any $\sigma\in\mathrm{Gal}(K/\mathbb{Q})$ must have $$\sigma(\zeta)=\zeta^k\text{ for some }1\leq k\leq 4,\qquad\quad \sigma(\sqrt[5]{5})=\zeta^r\sqrt[5]{5}\text{ for some }0\leq r\leq 4$$ (These comprise the $20$ different elements of $\mathrm{Gal}(K/\mathbb{Q})$.)

To understand $\mathrm{Gal}(K/\mathbb{Q})$, try to write it using a presentation (generators and relations). Can you think of any particular elements of $\mathrm{Gal}(K/\mathbb{Q})$ that, taken together, generate the entire group? Try to treat the two generators of the field "orthogonally" - find one automorphism that permutes the various powers of $\zeta$ and ignores $\sqrt[5]{5}$, and another automorphism that permutes the various 5th roots of $5$ and ignores any $\zeta$s (or at least, any $\zeta$s that aren't part of one of the 5th roots of $5$). Then combinations of these two should allow you to produce any desired effect on both $\zeta$ and $\sqrt[5]{5}$.

Then, determine the relations between those generating elements. That should make it easier to figure out the subgroups of $\mathrm{Gal}(K/\mathbb{Q})$ (and hence, the subfields of $K$).

Related Question