Transcendental element and algebraic closure

Question

I am trying to solve the following problem:

Let $\alpha$ be a transcendental element in some extension field of $H$. If $\frac{f(\alpha)}{g(\alpha)} \in H(\alpha)$ is algebraic over $H$ then this quotient is a constant in $H$.

I know that this is the same (or at least very similar) to prove that the algebraic closure of $H(\alpha)$ in $H$ is $H$. What did I?

Since $\frac{f(\alpha)}{g(\alpha)}$ is algebraic over $H$ then there exist some polinomial $h \in H[x]$ such that $h(\frac{f(\alpha)}{g(\alpha)})=0$.

Then if we multiply this by $g(\alpha)^n$ where $n$ is the degree of $h$ we can conclude that $g(\alpha)$ divides $f(\alpha)^n$ and $f(\alpha)$ divides $g(\alpha)^n$. Then, I know that somehow I can conclude from this that $g|f$ and $f|g$ and by this that they are associates and therefore $f(\alpha)=cg(\alpha)$ for some constant in $H$. But I need help in two things:

1. I know that somehow I am using that $H[x]$ is a Princial Ideal Domain (or an Unique Factorization Domain) to conclude that any irreducible factor of $g$ divides $f$ and vice-versa. But I dont understand clearly the argument, so if someone can help me I would be very grateful.

2. I cannot see why we need the hyphotesis that $\alpha$ is transcendental over $H$. Am I forgetting something?

Thank you everybody in advance!

Answer 1

Suppose that we have a rational function $f(X) = p(X)/q(X)$, written in lowest terms with $p(X),q(X)\in H[X]$, and suppose that $f$ satisfies a polynomial equation:

$$a_n (f(X))^n + a_{n-1} (f(X))^{n-1} \cdots + a_1 (f(X)) + a_0 = 0$$

Here $a_i \in H$. Clearing denominators:

$$a_n (p(X))^n + a_{n-1} (p(X))^{n-1} q(X) + \cdots + a_1 p(X) (q(X))^{n-1} + a_0 (q(X))^{n-1} = 0$$

From this equation, we can see that $q(X)$ is a factor of $a_n (p(X))^n$. Since $p$ and $q$ are relatively prime, $q(X)$ is a factor of $a_n$, and the only way this can happen is if $a_n=0$, or $q$ is a constant.

If $a_n=0$, it follows that $f$ cannot satisfy any nontrivial polynomial equation, and is therefore transcendental over $H$. If $q$ is a constant, then $f$ is a polynomial and it is easy to see it is transcendental.

Facts we need to make this work:

• It is possible to write a rational function in lowest terms. This follows from the existence of greatest common divisors in $H[X]$.
• If $p$ and $q$ are relatively prime, then $p^n$ and $q$ are relatively prime. This follows from unique factorization in $H[X]$, but it also follows from the simpler Bézout's identity: if $a$ and $b$ are polynomials such that $ap + bq=1$, then taking the n-th powers gives a linear combination of $p^n$ and $q$ that equals $1$.