I have just started out proofs so I am not fully grasping this concept yet.
Question: Is the set $\mathbb{R}^2$, with addition and multiplication defined below a field? Explain.
$(a, b) + (c, d) = (a + c, b + d)$
$(a, b) (c, d) = (ac, bd)$
Attempted solution:
We know: $f\colon F \times F \rightarrow F$
$$f(x, y) = x + y$$
$g\colon F \times F \rightarrow F$
$$g(x, y) = xy$$
So solving 1:
$$(a, b) + (c, d) = (a + c, b + d)$$
$$ \begin{align*} f(a, b) + (c, d) &= (a + b) + (c + d) &&\text{definition of field} \\ &= (a + c) + (b + d) \\ &= (a + c, b + d) \end{align*} $$ The second part I did almost the same thing using $g(x, y) = xy$.
Sorry for the formatting, I still don't know how to use MathJaX but any help is appreciated. If you can also explain for a beginner that would be great, thanks.