**Lemma 1:** If $P(x)$ is an integer polynomial with root $r$, then $P(x^2)$ is an integer polynomial with root $\sqrt{r}$.

**Lemma 2:** If $P(x)$ is an integer polynomial with root $r$, then $P(\sqrt{x})P(-\sqrt{x})$ is an integer polynomial with root $r^2$.

**Lemma 3:** If $P(x)$ is an integer polynomial with root $r$, then $P(x - t)$ is an integer polynomial with root $r + t$, for $t$ an integer.

**Lemma 4:** If $P, Q$ are integer polynomials with roots $r, s$, then an integer polynomial with root $r + s$ is given by the characteristic polynomial of $A \otimes I + I \otimes B$ where $A, B$ are the companion matrices of $P, Q$ and $\otimes$ denotes the Kronecker product. Multiplication by $r + s$ defines a $\mathbb{Q}$-linear transformation on $\mathbb{Q}[r, s]$, which has $\mathbb{Q}$-basis $r^i s^j$ where $i, j$ range from $0$ to one less than the degrees of $P$ and $q$, and the matrix above is the matrix of this linear transformation in that basis.

So apply Lemma 2 twice, then Lemma 3, then Lemma 4, then Lemma 1.