polynomial with rational coefficient have$ -\sqrt{2}$ as it minimal value?

Question

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polynomial with rational coefficients have $-\sqrt{2}$ as it minimal value ?

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Answer 1

The polynomial $\frac14(\frac14 x^4-x^3-x^2+6x+1)$ has $-\sqrt2$ as a minimum at $x=-\sqrt2$.

This is I found it: The minimum needs to be attained at an irrational value, so I set out to find a polynomial where it is attained at $\pm\sqrt2$. Thus I wanted $x^2-2$ to be a factor of $f'$, but I also want $f'$ to have an odd degree so that $f$ has a minimum. Thus I tried $f'(x)=\alpha(x^2-2)(x-q)$ for some rational $\alpha,q$, integrated and added an arbitrary rational constant $r$. Choosing appropriate constants yields the desired minimum at $-\sqrt2$.