# Field of zero characteristics

• I
I am interested in the following theorem:
Every field of zero characteristics has a prime subfield isomorphic to ℚ.
I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the multiplicative rule for this homomorphism, but I am not sure how to prove the additive. I am also interested in what does the notation
(a*1)(b*1)-1 exactly mean. Thanks.

## Answers and Replies

fresh_42
Mentor
I am interested in the following theorem:
Every field of zero characteristics has a prime subfield isomorphic to ℚ.
I am following the usual proof, where we identify every p∈ℚ as a/b , a∈ℤ,beℕ, and define h:ℚ→P as h(a/b)=(a*1)(b*1)-1 (where a*1=1+1+1... a times) I have worked out the multiplicative rule for this homomorphism, but I am not sure how to prove the additive. I am also interested in what does the notation
(a*1)(b*1)-1 exactly mean. Thanks.
All we know about the prime field is its characteristic and ##0,1 \in \mathbb{P}##. ##a\in \mathbb{P}## is not sure. However ##\underbrace{1+\ldots +1}_{a \text{ times}}=a \,\cdot \, 1## for ##a \in \mathbb{N}## is, and likewise for negative ##a##. That's why ##a## times ##1## is used instead of ##a##. We only have isomorphic images of ##a## in ##\mathbb{P}##.

For the homomorphism, I guess ##\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}## will be needed.

mathwonk
Science Advisor
Homework Helper
2020 Award
this probably won't help much until you have some commutative algebra but it gives the outline. First of all since the field F is an abelian group, there is exactly one additive homomorphism from the integers Z into F for each choice of the image of 1 from Z, so choose that image to be the unit element 1 in P. Now you have the unique additive homomorphism from Z to P that fresh described sending each positive integer n to 1+...+1 (n times). By definition of the multiplication in Z this is also a multiplicative map hence a ring map. Then since each non zero integer goes to an invertible element of F, (because F has characteristic zero), there is a unique extension of the previously defined ring homomorphism Z-->F to a ring homomorphism Q --> F. Then since Q is a field, this ring map is necessarily injective, hence defines an isomorphism onto the smallest subfield of F, i.e. the prime field.