# 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.

fresh_42
Mentor
2021 Award
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