Lower bound on the dimension of the field of rational functions

Let be a field and be its field of rational functions in indeterminate . Then the -dimension of is at least the cardinality of . #m/thm/ring

In particular, the following set linearly independent:

Proof

Let

where

then

which is zero iff . But this is impossible since .

#state/tidy | #lang/en | #SemBr