$\DeclareMathOperator{\Frac}{Frac}$ Hello everybody. My question is about Proposition $6.2.7.$ in the book "Basic Structures of Function Field Arithmetic" by David Goss.
Given a field $L$ with an $\sigma$ automorphism of infinite order, one can consider the ring $L\{\sigma\}$ that consists of all polynomial-like expressions $\sum_{n=0}^N a_n \cdot \sigma^n$ with multiplication given by $(a\cdot \sigma )\cdot (b \cdot \sigma):=a\cdot \sigma(b) \sigma^2$ etc.
This ring is graded by the degree $\deg(\sum_{n=0}^N a_n\cdot \sigma^n):=N,a_N\not =0$.
Let $R$ be a commutative subring of $L\{\sigma\}$.
The proposition is about proving that $R$ and $\mathscr{R}:=\bigoplus_{d=0}^\infty R_d$ are finitely generated $R_0$-algebras where $R_d:=\{f\in R\mid \deg(f)\leq d\}$.
In the proof it is claimed, that $\Frac(R)$ ($L\{\sigma\}$ is a domain) contains an element $\alpha$ of order $t:=\gcd(\deg(r)\mid r\in R)$.
I don't see how to prove this.
My attempt is to consider $\frac{\alpha}{\beta}\in \Frac(R)$ of degrees $\deg(\alpha)=a t$ and $\deg(\beta)=b t$. So we need to find $\alpha,\beta\in R$ such that $a-b=1$. Actually it would be enough to find $\alpha,\beta$ such that $\gcd(a,b)=1$, but I don't see how to show this either. Of course, the plain definition of $t$ does not imply the existence of an element in $R$ of degree $t$, but that such an element exists in the quotient field sound plausible, so I belive the statement of the proof must be correct.
The sequel of the proof is also a bit incomprehensive for me, but this is another topic.
I help somebody who can figure out the answer is willing to explain it tome. Thank You for Your effort!
greetings.