Let K be field and L be a subfield prove that

Question

Let $K$ be field and $L$ be a subfield prove that

A) if $K= \mathbb{R}$ and $\sqrt{2} \in L$ then $\mathbb{Q}(\sqrt2) \subset L$

B) If $K= \mathbb{Z}_p$ then $L=\mathbb{Z}_p$

I don't know how proving things like this looks like please help

Answer 1

Hints:

• Every field of characteristic zero has the rationals as a prime field, meaning: any field of characteristic contains an isomorphic copy of the rationals. This already solves (A) .

• In a positive characteristic $\;p>0\;,\;\;p\;$ a prime, the prime field is $\;\Bbb F_p:=\Bbb Z/p\Bbb Z\;$

Answer 2

Do this by definition. $\mathbb{Q}(\sqrt{2})$ is by definition the intersection of all subfields that contain $\mathbb{Q}$ and $\sqrt{2}$. So, to prove that $\mathbb{Q}(\sqrt{2}) \subseteq L$, it is enough to establish two things: (1) $\mathbb{Q} \subseteq L$ and (2) $\sqrt{2} \in L$.

Fact (2) is given to you for free. It remains to prove fact (1): $\mathbb{Q} \subseteq L$. Can you find a way to do this? This fact is very, very commonly used.