How to prove that the sum and product of two algebraic numbers is algebraic? Suppose $E/F$ is a field extension and $\alpha, \beta \in E$ are algebraic over $F$. Then it is not too hard to see that when $\alpha$ is nonzero, $1/\alpha$ is also algebraic. If $a_0 + a_1\alpha + \cdots + a_n \alpha^n = 0$, then dividing by $\alpha^{n}$ gives $$a_0\frac{1}{\alpha^n} + a_1\frac{1}{\alpha^{n-1}} + \cdots + a_n = 0.$$

Is there a similar elementary way to show that $\alpha + \beta$ and $\alpha \beta$ are also algebraic (i.e. finding an explicit formula for a polynomial that has $\alpha + \beta$ or $\alpha\beta$ as its root)? 

The only proof I know for this fact is the one where you show that $F(\alpha, \beta) / F$ is a finite field extension and thus an algebraic extension.
 A: Technically, you could find the automorphisms of the Galois closure of $F(\alpha,\beta)$ over $F$ (assuming this extension is separable) and compute the polynomial
$$
\prod_{\sigma \in \mathrm{Gal}}(x- \sigma(\alpha+\beta))
$$
or the same with $\alpha \beta$, but I don't believe this is what you are looking for. Since you can define Galois closures without knowing that $\alpha + \beta$ and $\alpha \beta$ are also algebraic, it is a legitimate way of proving it, but not a practical nor pedagogical one. 
Hope that helps,
A: The relevant construction is the Resultant of two polynomials. If $x$ and $y$ are algebraic and $P(x) = Q(y) = 0$ and $\deg Q=n$ then $z=x+y$ is a root of the resultant of $P(x)$ and $Q(z-x)$ (where we take this resultant by regarding $Q$ as a polynomial in only $x$) and $t=xy$ is a root of the resultant of $P(x)$ and $x^n Q(t/x).$
A: Okay, I'm giving a second answer because this one is clearly distinct from the first one. Recall that finding a polynomial over which $\alpha+\beta$ or $\alpha \beta$ is a root of $p(x) \in F[x]$ is equivalent to finding the eigenvalue of a square matrix over $F$ (living in some algebraic extension of $F$), since you can link the polynomial $p(x)$ to the companion matrix $C(p(x))$ which has precisely characteristic polynomial $p(x)$, hence the eigenvalues of the companion matrix are the roots of $p(x)$.
If $\alpha$ is an eigenvalue of $A$ with eigenvector $x \in V$ and $\beta$ is an eigenvalue of $B$ with eigenvector $y \in W$, then using the tensor product of $V$ and $W$, namely $V \otimes W$, we can compute
$$
(A \otimes I + I \otimes B)(x \otimes y) = (Ax \otimes y) + (x \otimes By) = (\alpha x \otimes y) + (x \otimes \beta y) = (\alpha + \beta) (x \otimes y)
$$
so that $\alpha + \beta$ is an eigenvalue of $A \otimes I + I \otimes B$. Also,
$$
(A \otimes B)(x \otimes y) = (Ax \otimes By) = (\alpha x \otimes \beta y) = \alpha \beta (x \otimes y)
$$
hence $\alpha \beta$ is an eigenvalue of the matrix $A \otimes B$. If you want explicit expressions for the polynomials you are looking for, you can just compute the characteristic polynomial of the tensor products.
Hope that helps,
A: Consider fields $ E \supseteq F $, and elements $ \alpha, \beta \in E $ algebraic over $ F $. We want to show $ \alpha + \beta $, $ \alpha \beta $ are algebraic over $ F $ too. If even one of $ \alpha, \beta $ are $ 0 $, the result is trivial, so let's take both $ \alpha, \beta $ to be non-zero.
We have $ \alpha ^m + a_{m-1} \alpha ^{m-1} + \ldots + a_0 = 0 $ ( each $ a_i \in F $ ), and $ \beta ^n + b_{n-1} \beta ^{n-1} + \ldots + b_0 = 0 $ ( each $ b_j \in F $ ).
(The first equation lets us express all powers of $ \alpha $ as $F$-combinations of $ 1, \alpha, \ldots, \alpha ^{m-1} $. Similarly for $ \beta $)
Let $$ Z := \, [ \, \alpha ^0 \beta ^0, \alpha ^0 \beta ^1, \ldots, \alpha ^0 \beta ^{n-1} ; \alpha ^1 \beta ^0, \ldots, \alpha ^1 \beta ^{n-1} ; \ldots ; \alpha ^{m-1} \beta ^{0}, \ldots, \alpha ^{m-1} \beta ^{n-1} \, ]^{T} \in E^{mn} $$
Now notice we can express $ (\alpha + \beta)Z $ as $ M_1 Z $ with $ M_1 \in F^{mn \times mn} $. So $ ( ( \alpha + \beta ) I - M_1 ) Z = 0 $, and as $ Z \neq 0 $ we have $ \det( (\alpha + \beta)I - M_1 ) = 0 $. Hence $ \alpha + \beta $ is a root of the polynomial $ P(t) := \det( tI - M_1 ) \in F[t] $, and is therefore algebraic over $ F $. Similarly we can show $ \alpha \beta $ is algebraic over $ F $ (Write $ \alpha \beta Z = M_2 Z $ and proceed as above).
A: Let $\alpha$ have minimal polynomial $p(x)$ and let $\beta$ have minimal polynomial $q(x)$. Then $V = F[x, y]/(p(x), q(y))$ is a finite-dimensional vector space over $F$ of dimension $\deg p \deg q$ (it is not necessarily the same dimension as $F(\alpha, \beta)$, for example when $\alpha = \beta$); moreover, it has an explicit basis
$$x^i y^j : 0 \le i < \deg p, 0 \le j < \deg q.$$
$xy$ and $x + y$ act by left multiplication on $V$ and one can write down explicit matrices for this action in the basis above in terms of the coefficients of $p$ and $q$. Now apply the Cayley-Hamilton theorem. 
This argument proves the stronger result that if $F$ is the fraction field of some domain $D$ and $\alpha, \beta$ are integral over $D$ (hence $p, q$ are monic with coefficients in $D$) then so are $\alpha \beta, \alpha + \beta$. 
