I haven't been able to find any counterexamples for either of the two. (1) seemed intuitively true but I had my doubts on (2) and couldn't find one. If there aren't any counterexamples, how can I go about proving them? A full solution isn't necessary. I'm having trouble figuring out what is the correct way in which to begin/outline the proof.
Note that subfield is used interchangeably with extension field.
Prove or give a counterexample:
(1) If $A$ and $B$ are subfields of a field $F$, then $\{b+a|b\in B, a\in A\}$ is also a subfield of $F$.
Also, (2) If $A$ and $B$ are subfields of a field $F$, then $\{ba|b\in B, a\in A\}$ is also a subfield of $F$.
I know that in order for something to be considered a subfield it must satisfy; having at least two elements, must have addition and multiplication, and add/mult inverses. I can't seem to make the connection between these conditions and the generic circumstances I'm being asked to prove.
Any help would be greatly appreciated.
What you must do is check if the subset satisfies the subfield axioms. So for example for the first subset, if you want to check the closedness under addition, you take $x,x' \in A$, $y,y' \in B$. Then $x+y$ and $x'+y'$ are in the subset $X = \{ b+a \mid b \in B, a \in A \}$, so their sum must be in $X$ again. But their sum is $(x+y)+(x'+y') = (x+x')+(y+y')$, and since $A$ and $B$ are subfields, $x+x' \in A$ and $y+y' \in B$. This proves $X$ is closed under addition.
Now let's see. Does it look like the subset in question (1) is closed under multiplication? Not really. You should find an explicit counterexample. Consider the following two subfields of $\mathbb{R}(x,y)$: $A = \mathbb{R}(x)$ and $B = \mathbb{R}(x)$. Then you cannot write $xy$ as the sum of a polynomial in $x$ and a polynomial in $y$. Indeed suppose $xy = P(x) + Q(y)$, then $P(0) + Q(y) = 0 \cdot y = 0$ for all $y$, thus $Q$ is constant. By a similar reasoning $P$ is constant too, which is absurd.
For (2), it's not closed under addition. Take the same $A$ and $B$, then you cannot write $x + y$ as the product of a polynomial in $x$ and a polynomial in $y$ (by a similar reasoning as before.