Isomorphism between a polynomial field and $\mathbb{R}$

Question

Let $$\mathbb{K} = \mathbb{Q}[x]/(x^3-2)$$ Show that exists only one isomorphism between $\mathbb{K}$ and a subfield of $\mathbb{R}$ and find the subfield.

I tried to find the subfield but i got no results. Can I have just some tips?

Answer 1

Hint: Intuitively, in the quotient we get $X=2^{1/3}$. Thus the quotient ring are polynomials in the cube root of $2$. This suggests that the quotient ring is $\Bbb{Q}(2^{1/3})$. Can you make this formal?

Answer 2

One way to do that in general is as follow.

1. You can verificate that your polynomial $x^3-2$ is irreducible for Eisenstein's criterion.
2. Take $\alpha$ a only one real root of your polynomial and defined $\varphi:\mathbb{Q}[x]\rightarrow \mathbb{Q}(\alpha)$ by $q(x)\mapsto q(\alpha)$ the evaluation homomorphism which is onto.
3. $ker(\varphi)=\{q(x)\in \mathbb{Q}[x]: q(\alpha)=0\}$, and as $\mathbb{Q}[x]$ is a PID then you should check that is iqual to ideal generate for $x^3-2$.
4. Finally, you can use the isomorphism theorem to see that $\mathbb{Q}[x]/(x^3-2)\cong \mathbb{Q}(\alpha)\subset \mathbb{R}$.