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Reading Neukirch's Algebraic Number Theory I came across the concepts of global and local fields—the first being defined as "a finite extension of either $\mathbb{Q}$ or of $\mathbb{F}_p(t)$ for a prime number $p$" and the second definition being equivalent to "a finite extension of either $\mathbb{Q}_p$ or of $\mathbb{F}_p((t))$ for a prime number $p$".

In other books and various sources online I saw some other definitions for the case of positive characteristics, for example in Wikipedia a global field of positive characteristics is defined as a finite extension of $\mathbb{F}_q(t)$ for some prime power $q=p^n$ and a local field of positive characteristics is just a field of the form $\mathbb{F}_q((t))$ for some $q=p^n$ (not a finite extension thereof).

I'm very confused and my questions are:

  • Are the finite extensions of $\mathbb{F}_p(t)$ precisely the finite extensions of $\mathbb{F}_q(t)$ for all powers $q=p^n$?

  • Are the finite extensions of $\mathbb{F}_p((t))$ precisely the fields $\mathbb{F}_q((t))$ for all powers $q=p^n$ or precisely the finite extensions of $\mathbb{F}_q((t))$ for all powers $q=p^n$?

Or maybe different definitions are being used?

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