Let K be a field with characteristic p > 0 and M = K ( X ,

Dayanara Terry 2022-06-29 Answered
Let K be a field with characteristic p > 0 and M = K ( X , Y ) the field of rational functions in 2 variables over K. We consider the subfield L = K ( X p , Y p ) M.
Show that [ M : L ] = p 2 .
I guess I need the property that [ K ( x ) : K ( x n ) ] = n, which we showed already. But I actually do not know how to use it here. I can not work well with that field in 2 variables..
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Answers (1)

lywiau63
Answered 2022-06-30 Author has 13 answers
Recall that for some field J so that L J M you have that the degree of the extension L M is the product of the degrees of the extensions L J and J M.
Use this for example with J = K ( X p , Y ), applying the result you know twice.
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