# Definition of local and global fields

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?