/sci/ - Science & Math

Doesn't the statement

"[math]k(X)[/math] is a finite extension of the field [math]f*(k(\mathbb{A}^m))\cong k(t_1,...,t_m)[/math], hence [math]k(X)[/math] has transcendence degree m over k"

imply that the transcendence degree of [math]k(X)[/math] is AT LEAST m? how do we know that the finite extension that they mention doesn't introduce any algebraically independent elements?

Is it because "A finite extension of B" implies "A integral over B" implies "A is algebraically dependent on B"?

Also, how do we know that the [math]t_1,...,t_m[/math] from this statement (which I believe are the image of each coordinate function [math]x_i[/math] of [math]\mathbb{A}^m[/math] under the homomorphism [math]f^*[/math]) are algebraically independent?

My best explanation is that it's because the image of [math]k(\mathbb{A}^m)[/math] to be an ideal of [math]k(X)[/math], hence implies a subvariety of X, which can't happen since X is irreducible.