Expressing a number field in a simpler way 
Express $\mathbb{Q}(\sqrt{3},\sqrt[3]{5})$ in the form $\mathbb{Q}(\theta)$.
Hint: Let $f,g$ be the minimal polynomials of $\sqrt{3}$ and $\sqrt[3]{5}$, respectively. Factor $f$ and $g$ in $\mathbb{C}$ to obtain expressions of the form $$f(x)=\prod\limits_{j=1}^n (x-\alpha_j),g(x)=\prod\limits_{k=1}^m (x-\beta_k)$$
where $\alpha_1=\sqrt{3}$ and $\beta_1=\sqrt[3]{5}$. Choose $c \in \mathbb{Q}$\ $\lbrace 0 \rbrace$ so that $$\alpha+c\beta \notin \lbrace \alpha_j+c\beta_k|1\leq j \leq n, 2 \leq k \leq m \rbrace$$ It can then be shown that $\mathbb{Q}(\sqrt{3},\sqrt[3]{5})=\mathbb{Q}(\sqrt{3}+c\sqrt[3]{5})$

Following the hint, I think $c=1$ works. Now I try to prove that $\mathbb{Q}(\sqrt{3},\sqrt[3]{5}) \subseteq \mathbb{Q}(\sqrt{3}+\sqrt[3]{5})$ (the other containment is easy), but I'm having trouble with this last step.

Denote
$$
a=\sqrt{3},\quad b=\sqrt[3]{5}, \quad x = a+b.
$$
Let's express $1,x,x^2,x^3,x^4,x^5$ as linear combinations of $1,a,b,ab,b^2,ab^2$:
$$
\begin{array}{lll}
1= 1; & \\
x=  a+b; & \\
x^2= (a+b)^2 & = a^2+2ab+b^2 & = 3+2ab+b^2;\\
x^3= (a+b)^3 & = a^3+3a^2b+3ab^2+b^3 & = 3a+9b+3ab^2+5;\\
x^4= (a+b)^4 & = a^4+4a^3b+6a^2b^2+4ab^3+b^4 & = 9+12ab+18b^2+20a+5b;\\
x^5= (a+b)^5 & = a^5+5a^4b+10a^3b^2+10a^2b^3+5ab^4+b^5 & = 9a+45b+30ab^2+150+25ab+5b^2.
\end{array}
$$
Following this way, one can express:
$$
\left(
\begin{array}{c}
1 \\ x \\ x^2 \\ x^3 \\ x^4 \\ x^5 
\end{array}
\right)
=
\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & 1 & 1 & 0 & 0 & 0 \\
3 & 0 & 0 & 2 & 1 & 0 \\
5 & 3 & 9 & 0 & 0 & 3 \\
9 & 20 & 5 & 12 & 18 & 0 \\
150 & 9 & 45 & 25 & 5 & 30
\end{array}
\right)
\cdot
\left(
\begin{array}{c}
1 \\ a \\ b \\ ab \\ b^2 \\ ab^2 
\end{array}
\right),
$$
or, say,
$$
X = M\cdot A.
$$
From this,
$$
A = M^{-1}\cdot X;
$$
in particular,
$$
\sqrt{3} = a = \frac{1}{2403}\left(-4095+3015x-1170x^2-720x^3+45x^4+72x^5\right),
$$
$$
\sqrt[3]{5} = b = \frac{1}{2403}\left(4095-612x+1170x^2+720x^3-45x^4-72x^5\right),
$$
(where $x=\sqrt{3}+\sqrt[3]{5}$).
