Degree of a field extension by a transcendental element

The natural $F$-basis of $F(x)$ is $$\{ x^k, k\ge 0\} \cup \{ x^l/h^m, m\ge 1,l<\deg(h), h \in F[x]\text{ monic irreducible}\}$$ Thus (for $F$ infinite) the cardinality of the basis is comprised between that of $F$ and $F[x]^2$, ie. it is the same as $F$.

Tags:

Abstract Algebra

Cardinals

Field Theory

Transcendental Numbers

Extension Field

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