Irreducibility of $2x^3-3x^2+6$ in $\mathbb Q[x]$

Question

The following statement is given, i need to check whether its true\false.

There exists a subfield $F$ of $\mathbb{C}$ such that $F\not\subseteq\mathbb R$ and $$F \cong \mathbb Q[X]/(2X^3 − 3X^2 + 6).$$

My attempt:

I thought since its cubic polynomial it will have a real solution, how do I show its not in $\mathbb Q$, and then the fact that $F\not\subseteq\mathbb R$?

Ans - Eisenstein and Rational root test, settle the first part. How to show that $F\not\subseteq\mathbb R$.

I know for a fact that this is a true statement.

Answer 1

Hint 1. A rational root $p/q$ of $f(x):=2x^3-3x^2+6$ with $\gcd(p,q)=1$ is such that $p$ divides $6$ and $q$ divides $2$ (See the Rational Root Theorem).

Hint 2. Note that $f'(x)=6x(x-1)$, $x=0$ is a local maximum and $x=1$ is a local minimum with $f(1)=5>0$. Show that from this it follows that the roots are not all real.

Answer 2

Hint: Rational root theorem. $\hphantom{imaginary space}$

Answer 3

This follows from Eisensteins's criterion, with $p=3$.