How do I find a splitting field $x^8-3$ over $\mathbb{Q}$?

Question

Here's the situation. I am in this algebra class, and so far we have defined splitting fields and proved their existence and uniqueness. We have not yet decided on any rigorous definition of complex numbers, by the way. For a homework question (and yes, we are allowed to use any internet resources we want),

I have to find a splitting field for $x^8-3$ over $\mathbb{Q}$ and find its degree of extension.

I don't really know how to go about it. If I can use that the complex numbers are algebraically closed, can't I just adjoin all of the roots, or is there something more explicit that I can do in order to find the degree of extension?

Answer 1

Denote by $K$ the splitting field of $x^8-1$. Clearly $x^8-1$ has $8$ complex roots, namely $\zeta^i\cdot\sqrt[8]{3}$ for $i=0,\dots,7$ and $\zeta$ is a primitive $8^{th}$ root of unity. Since $K$ contains $\zeta\cdot\sqrt[8]{3}$ and $\sqrt[8]{3}$, it must contain their quotient $\zeta$. It also contains $\sqrt[8]{3}$, and we can conclude $K\supseteq\mathbb{Q}(\zeta,\sqrt[8]{3})$. On the other hand, any field containing these two elements must contain the splitting field for $x^8-1$, hence $K=\mathbb{Q}(\zeta,\sqrt[8]{3})$.

To determine the dimension of this extension, recall that field automorphisms permute the roots of minimal polynomials. $\sqrt[8]{3}$ has minimal polynomial $x^8-3$ over $\mathbb{Q}$, ($x^8-3$ is 3-Eisenstein, hence irreducible). $\zeta$ has minimal polynomial $\Phi_8(x)=x^4+1$ (see below), hence $\mathbb{Q}(\zeta)$ is an extension of dimension 4 over $\mathbb{Q}$. It is not difficult to see that $\zeta\notin\mathbb{Q}(\sqrt[8]{3})$, hence $K$ is of dimension $4\cdot8=32$ over $\mathbb{Q}$.

Last, recall (see Dummit & Foote 13.6, for instance) that the $n^{th}$ roots of unity satisfy

$x^n-1=\prod\Phi_d(x)$

where $d$ goes over all the divisors of $n$. In our case, $n=8$,

$x^8-1 = (x^4-1)\cdot(x^4+1)=(x-1)(x+1)(x^2+1)\cdot(x^4+1)$

where the first three multipliers at the last equality above are $\Phi_1(x)=x-1$, $\Phi_2(x)=x+1$ and $\Phi_4(x)=x^2+1$.

As a final remark, we could have taken $\zeta=(1+i)\sqrt{2}/2$. From a similar argument to before, $\mathbb{Q}(\sqrt{2},i)\supseteq\mathbb{Q}(\zeta)$. However, both sides are a 4-dimensional extension over $\mathbb{Q}$, hence they must be equal. This can also be proven directly, by writing $i$ and $\sqrt{2}$ only using powers of $\zeta$.

To summarize, the splitting field of $x^8-3$ is $\mathbb{Q}(\sqrt{2},i,\sqrt[8]{3})=\mathbb{Q}(\zeta,\sqrt[8]{3})$.

Answer 2

Just saying "adjoin all roots" isn't really going to help in figuring out the degree of the extension.

So you really want to be more explicit.

One root is clearly $\sqrt[8]{3}$. Another is $-\sqrt[8]{3}$; but these are the only two real roots, and the splitting field is supposed to have 8 roots, so they are not it; you'll certainly need to deal with complex numbers. But you should know what the degree of $\mathbb{Q}[\sqrt[8]{3}]$ over $\mathbb{Q}$ is, which will also get you started with the degrees.

How much do you know about "complex roots of unity"? You want some complex number $\zeta$ with the property that $\zeta^8 = 1$. Look up "primitive roots of unity". That should help you continue.