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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.