I need to show that the prime field $\displaystyle \mathbb{R}$ and the prime field $\displaystyle \mathbb{C}$ are in $\displaystyle \mathbb{Q}$
could i get help showing one of these (the harder one) then ill try doing the other one... ??
I assume you mean the prime subfield of $\displaystyle \mathbb{R}$, notice that by definition this is the intersection of all subfields of $\displaystyle \mathbb{R}$ ie. $\displaystyle P(\mathbb{R})= \bigcap_{B\subset \mathbb{R} : \ B \ field } B$. Since $\displaystyle 1 \in P(\mathbb{R})$ and it is a field, it contains a copy of $\displaystyle \mathbb{Z}$. Now $\displaystyle \mathbb{Q}$ is the field of fractions of $\displaystyle \mathbb{Z}$ so $\displaystyle \mathbb{Q}$ is the smallest field containing $\displaystyle \mathbb{Z}$, so $\displaystyle \mathbb{Q} \subset P(\mathbb{R} )$ and since $\displaystyle P(\mathbb{R}) \cap \mathbb{Q} = P(\mathbb{R})$...
wow i was completely off on the question.. sorry.
i) Show that every subfield of $\displaystyle \mathbb{C}$ contains $\displaystyle \mathbb{Q}$
I thought $\displaystyle \mathbb{Z}[i] $ was a subfield of $\displaystyle \mathbb{C}$ but there is no $\displaystyle \mathbb{Q}$ subfield of $\displaystyle \mathbb{Z}[i]$... but wouldn't that disprove this statement... $\displaystyle (\mathbb{Z}[i]$ units are 1,-1,i,-i )
ii) Show that the prime field of $\displaystyle \mathbb{R}$ is $\displaystyle \mathbb{Q}$
iii) Show that the prime field of $\displaystyle \mathbb{C}$ is $\displaystyle \mathbb{Q}$
Read carefully my first post and you'll notice that it answers all three questions.
As for $\displaystyle \mathbb{Z} [i]$ this is, by definition, the smallest ring that contains both $\displaystyle \mathbb{Z}$ and $\displaystyle i$, and $\displaystyle \mathbb{Z} (i)$ is the aboves fraction field, and since it contains $\displaystyle \mathbb{Z}$ it must contain $\displaystyle \mathbb{Q}$
Ohhhh so what your saying in words is that
Because the intersection of the prime subfiled of $\displaystyle \mathbb{R} $ and $\displaystyle \mathbb{Q} $ is just the prime subfield of $\displaystyle \mathbb{R} $ , we can conclude that $\displaystyle \mathbb{Q} = P(R)$ ,
and the proof is the same thing for $\displaystyle \mathbb{C} $ to show the prime subfiled of $\displaystyle \mathbb{C} $ is $\displaystyle \mathbb{Q} $ ??? (basically just change the R's to C's)