High School Why does every subfield of Complex number have a copy of Q?

[fresh_42]

But werebn't we talking about subfields?

Yes, but we also got picky. $\mathbb{Z}_7$ is a field and $\{0,1,2,3,4,5,6\} \subset \mathbb{C}$. And yes it is no subfield, because it cannot be embedded as a field into $\mathbb{C}$. So in the same way you take this embedding for granted, @Buffu may take $0,1 \in F$, the additional closure or other axioms for granted, which reduces the entire exercise to: Obvious. I only wanted to say, that this property in the meaning / definition of subfield is used same as the axioms of a field are used.

 

[Buffu]

That is why I asked if you were supposed to be learning how to do proofs. We could take the entire thing for granted, but that is not a proof. We know that the axioms are true. If you are learning how to do math proofs, you have to refer to the things you know in writing where they apply and are used.

I will mention every axiom used from next time when writing proofs :).

 

[fresh_42]

I will mention every axiom used from next time when writing proofs :).

At least it will do no harm, except ...
Just a (not completely serious hint) in case you will turn in a paper for your master's degree or similar: Write it down in a way that you understand all steps. Make a copy for yourself. Then wipe out every second step and hand it over .

 

[FactChecker]

I will mention every axiom used from next time when writing proofs :).

Good. The first few years of proofs should include all the steps. You will have a lot of time to learn which steps to skip. A thesis can skip some steps and an article for publication must skip a lot.