I've heard of and seen some proofs that the product and sum of two algebraic numbers is algebraic, however many of them are quite complex and require a variety of machinery (from matrix eigenvalues to complex analysis).
Are there simpler proofs of the following?
A rational + an algebraic is algebraic.
A rational + a transcendental is transcendental.
Rational + algebraic = algebraic
Let $x$ be an algebraic number, so $x^n + a_1 x^{n-1} + \cdots + a_n = 0$ for rational numbers $a_1, \cdots, a_n$.
Then, $x+\frac pq$ satisfies the polynomial equation $\left(z-\frac pq\right)^n + a_1 \left(z-\frac pq\right)^{n-1} + \cdots + a_n\left(z-\frac pq\right) = 0$, so it is algebraic.
Rational + transcendental = transcendental
Let $x$ be transcendental.
If $x+\frac p q$ is algebraic, then so is $\left(x+\frac pq\right) - \frac p q = x$, contradiction. So we conclude that $x+\frac p q$ is transcendental.