- #1

- 237

- 5

## Homework Statement

Show that if [itex] \alpha[/itex] is real and has degree 10 over [itex]\mathbb{Q}[/itex] then

[itex]\mathbb{Q}[\alpha]=\mathbb{Q}[\alpha^3][/itex]

## Homework Equations

## The Attempt at a Solution

It is clear that [itex]\mathbb{Q}[\alpha^3]\subset \mathbb{Q}[\alpha][/itex]. This gives us the

sequence of fields [itex]\mathbb{Q}\subset \mathbb{Q}[\alpha^3]\subset \mathbb{Q}[\alpha][/itex]. Since these are finite extensions, we then have [itex][\mathbb{Q}[\alpha]:\mathbb{Q}]=10=[\mathbb{Q}[\alpha]:\mathbb{Q}[\alpha^3]][\mathbb{Q}[\alpha^3]:\mathbb{Q}][/itex].

Since [itex]x^3-\alpha^3\in \mathbb{Q}[\alpha^3][x][/itex] is of degree 3 and has [itex]\alpha[/itex] as a root, [itex][\mathbb{Q}[\alpha]:\mathbb{Q}[\alpha^3]]\leq 3[/itex]. Since it must also divide 10, it must be 1 or 2. The goal then is to show that it cannot be 2 (or then that [itex][\mathbb{Q}[\alpha^3]:\mathbb{Q}][/itex] cannot be 5.)

I have tried going further on this point, but I think I'm just getting stubborn and missing something subtle. Any hints would be appreciated.

-Theorem