Find a nonzero polynomial $f(X)$ with integer coefficients such that $f(\sqrt3 + \sqrt7)=0$ Find a nonzero polynomial $f(X)$ with integer coefficients such that $f(\sqrt3 + \sqrt7)=0$
I was thinking it might be possible using the rational roots theorem, but I've never used it with square roots, only integers. Any advice?
 A: $$\prod_{(s_1,s_2)\in\{-1,+1\}^2}\!\!\!\!\!\!\!(x-s_1\sqrt{3}-s_2\sqrt{7}) =\color{red}{x^4-20x^2+16}$$
is an element of $\mathbb{Z}[x]$ that clearly vanishes at $x=\sqrt{3}+\sqrt{7}$. Another approach for achieving the same: we may consider the ring $\mathbb{Q}[a,b]/(a^2-3,b^2-7)$ and the powers $(a+b)^k$ for $k\in\{0,1,2,3,4\}$ represented with respect to the base $1,a,b,ab$. We have five vectors in a vector space with dimension four, hence a non-trivial combination of them gives the zero vector. Such a combination is associated with a polynomial (actually, the minimal polynomial) with integer coefficients having $\sqrt{3}+\sqrt{7}$ as a root:
$$\begin{array}{|c|c|c|c|c|}\hline & 1 & a & b & ab \\ 
\hline (a+b)^0 & 1 & 0 & 0 & 0 \\ \hline (a+b)^1 & 0 & 1 & 1 & 0 \\
\hline (a+b)^2 & 10 & 0 & 0 & 2\\
\hline (a+b)^3 & 0 & 24 & 16 & 0 \\
\hline (a+b)^4 & 184 & 0 & 0 & 40 \\\hline\end{array}$$
we may notice, by Gaussian elimination, that the last row ($k=4$) minus 20 times the third row ($k=2$) plus 16 times the first row ($k=0$) equals zero. So we recover the polynomial $x^4-20x^2+16$ and by studying the rank of the last matrix we may also prove it is the minimal polynomial of $\sqrt{3}+\sqrt{7}$ over $\mathbb{Q}$.
A: A subtle variation of @dxiv's idea 
Let $x=\sqrt{3}+\sqrt{7} \Rightarrow x-\sqrt{3}=\sqrt{7}$
Now, squaring both sides, we get 
$$x^2+3-2\sqrt{3}x=7 \Rightarrow x^2-4=2\sqrt{3}x$$
Again, squaring both sides, we get
$$x^4-8x^2+16=12x^2 \Rightarrow x^4-20x^2+16=0$$
A: You can find the minimal polynomial of $x=\sqrt 3+\sqrt 7$ (you will find the same polynomial for the four possibilities $x=\pm\sqrt3\pm\sqrt7$ because of known conjugates of the first given $x$). This is squaring $$f(x)=x^4-20x^2+16$$
The answer of your question can be anyone of the infinitely many polynomials
$$f(x)g(x)$$ where $0\ne g\in \Bbb Z[x]$ (obviously $f$ has relevance!).
