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Let $K$ be a number field of degree 3. Show that $K$ is isomorphic to a quotient $\mathbb Q[x]/(f)$, with $f = x^3 + ax + b$ in $\mathbb Z[x]$ irreducible in $\mathbb Q[x]$ (without using the result that every number field is defined by some irreducible polynomial over $\mathbb Q$).

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Let $\alpha \in K \setminus \mathbb Q$ and note that $K=\mathbb Q(\alpha)$. Then we have that $K \cong \mathbb Q[x]/\min_\mathbb Q(\alpha,x)$. Now $f(x)=\min_{\mathbb Q}(\alpha,x)=x^3+bx^2+cx+d$, applying the automorphism of $\mathbb Q[x]$ that sends $x\mapsto x-b/3$ we find that $\mathbb Q[x]/g(x)$ where $g(x)=x^3+qx+p$ for some new rationals $q,p$. You may recognize this trick from the derivation of the cubic formula. Now you can check that a suitable integer multiple of $\alpha-b/3$ is an algebraic integer with the desired minimal polynomial. It's simply a matter of looking carefully at what you do when you prove that for any algebraic number $\alpha$ there exists a $k$ such that $k\alpha$ is an algebraic integer.

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