So presumably this question is very basic, but I'm having some trouble with apparent contradictions in my reasoning.
Let k be a field and k⊆K a field extension. We say that K is a finitely generated field extension if it is finitely generated over k as a k-algebra. We say that K is a finite field extension if it is finite dimensional as a k-vector space. By Zariski's lemma, these are equivalent concepts: A finitely generated field extension is finite.
We say that an element t∈K is transcendental over k if there is no monic polynomial with coefficients in k for which t is a root.
So far, is this correct? I think so. Which brings me to my confusion. I have encountered the term "finitely generated k-algebra of transcendence degree 1". I don't understand how such an extension can exist. If k⊆K is a field, and t∈K is a transcendental element over k, then the elements 1,t,t2,t3,t4,… would be algebraically independent. Indeed if there was a dependency, then t would fail to be transcendental. But then this is an infinite set of generators.
Where is the flaw in my reasoning? How can a transcendence degree 1 field extension be finitely generated?
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