# Minimal Polynomial of $\sqrt{2}+\sqrt{3}$ over $\mathbb C$ (Field Theory)

I know that the minimal polynomial of $$\sqrt{2}+\sqrt{3}$$ over $$\mathbb Q$$ is indeed $$x^4-10x^2+1$$ (as shown multiple times already) because of the fact that it is (1) monic and (2) irreducible in $$\mathbb Q$$.

Now i am asked to find the minimal polynomials for $$\sqrt{2}+\sqrt{3}$$ over $$\mathbb R$$ and $$\mathbb C$$ aswell.

I used the same technique as for $$\mathbb Q$$ but halted as soon as i reached the irreducible polynomial, which in this case is $$p(x)=x-\sqrt{2}-\sqrt{3}$$.

The polynomial $$p(x)$$ is the minimal polynomial of $$\sqrt{2}+\sqrt{3}$$ over $$\mathbb R$$ and $$\mathbb C$$. It should be trivial for $$\mathbb R$$ and $$\mathbb C$$ because $$p(x)$$ is indeed monic and irreducible by the definition of $$\mathbb R$$ and $$\mathbb C$$ (since all polynomials of $$deg(1)$$ are irreducible).

Is my thought process correct?

• Yes. It is also trivial because for any field $F$ and any $\alpha\in F$ the minimal polynomial of $\alpha$ is $X - \alpha.$ Jan 10 '20 at 19:26
• We may notice that $mathbb{C}$ is algebraically closed, so any polynomial splits in linear factors, so the minimal polynomial of any $\alpha\in\mathbb{C}$ is linear, i.e. is $x-\alpha$ Jan 10 '20 at 19:29
• @DietrichBurde Oh sure, it is just that i am quite new to field theory and the concept of minimal polynomials so i wanted to know if my idea is correct. It looked strange to me at first because $\sqrt{2}+\sqrt{3}$ is in $\mathbb R$ and from there on out we wanted to find a minimal polynomial in a class "above" so to say. But since $\mathbb R$ is in $\mathbb C$ i think my concerns were unnecessary. Thank you for all the answers! Jan 10 '20 at 19:48

You rightly noticed that $$\sqrt2+\sqrt3\in\Bbb R$$ (aswell as $$\sqrt2+\sqrt3\in\Bbb C$$). And, indeed, for every $$\alpha\in L$$, where $$L$$ is a field, the minimal polynomial of $$\alpha$$ over $$L$$ is just $$f(x)=x-\alpha\in L[x]$$. This is both (1) monic and (2) irreducible as you showed by yourself.
• A follow up question: Does that mean that technically speaking $p(x) = x - (\sqrt{2} + \sqrt{3})$ is also a minimal polynomial in $\mathbb Q$ but since $\sqrt{3}$ and $\sqrt{2}$ are not part of $\mathbb Q$ we first have to remove the square-roots? And would that imply since $p(x) = x - (\sqrt{2} + \sqrt{3})$ is a sort of minimal polynomial in $\mathbb Q$, we can be sure that $x^4−10x^2+1$ is a minimal polynomial in $\mathbb Q$? Jan 11 '20 at 9:10
• Yes, technically speaking $p(x)=x-(\sqrt2+\sqrt3)$ is a minimal polynomial, but not in $\Bbb Q[x]$ as the coefficients are not in $\Bbb Q$. That's why we have to remove the square roots by squaring until we obtain something in $\Bbb Q[x]$. Then, if you finnd a monic and irreducible polynomial with root $\sqrt2+\sqrt3$ as minimal polynomials are unique. Jan 11 '20 at 10:13