How to prove this: If $t$ is a transcendental number, then $5t^{4}+8t+3$ is also transcendental?

Question

Can I prove as follows? If $5t^{4}+8t+3$ is not transcendental, then $5t^{4}+8t+3$ is a solution of a polynomial $p$. If you expand $p(5t^{4}+8t+3)=0$ and write it in the form of another polynomial $p'(t)=0$, this indicates that $t$ is the solution of this polynomial, which contradicts the fact $t$ is transcendental.

Answer 1

Yes, it is as simple as that. The statement is equivalent to: if $5t^4 + 8t + 3$ is algebraic, then $t$ is algebraic. Your argument shows exactly that: given a polynomial $p(X)$ with $p(5t^4 + 8t + 3) = 0$, the polynomial $q(X) := p(5X^4 + 8X + 3)$ satisfies $q(t) = 0$.

Answer 2

Let $p(X)$ be any non-constant polynomial with rational coefficients and assume that $p(t)$ is an algebraic number. Then by definition there exists a non-constant polynomial $q(X)$ such that $q(p(t))=0$. But $q(p(X))$ is a non-constant polynomial with rational coefficients. Hence $t$ is algebraic.