- #26

- 2,012

- 1

I wish you would use the dot notation because I do not think that multiplication by theTake a nonzero element a in the algebraic closure. If pa != 0, then p*1 = pa*a^-1 != 0.

*element*p in the algebraic closure of of Z/pZ is necessarily the same as [itex] p \cdot a = (a +...+a) = \sum_{i=1}^p a[/itex]. I would use the asterick only for multiplication in the field.

So, you are saying that if [itex] p \cdot a \neq 0 [/itex], then [itex] p \cdot 1 = (p \cdot (a*a^{-1}) ) = (p \cdot a)* a^{-1} [/itex] where in the last step I used the distributive property. That is obviously impossible because 1 is in Z/pZ and we thus know its characteristic is p. So I guess what you did checks outs, but I still think you should always use the dot notation when dealing with characteristics of fields.

There is a theorem in my book that says: "A field is either of prime characteristic p and contains a subfield isomorphic to Z_p or of characteristic 0 and contains a subfield isomorphic to Q."But really, the easiest way to see this is to think about characteristic in terms of prime subfields. Since the prime subfield of the algebraic closure of Z/pZ is Z/pZ, we're done.

Simply containing a subfield isomorphic of Z_p is not enough to conclude that the field has prime characteristic p (at least from this theorem).