So I'm a first year just starting analysis and linear algebra and so far, I suck. I'm not sure how to understand, or more so, interpret everything.
So here are my questions:
1) So the elements of the cartesian product of two fields would be in the form $(x, y)$. So I'm confused. Somewhere I saw that $(1,0)$ is the multiplicative identity, but shouldn't $(1, 1)$ be a multiplicative identity? Why isn't $(0, 1)$ the multiplicative identity then?
2) So some Cartesian products of fields are not fields and some are. For example: the complex numbers, they are a field. However, the reason why some cartesian products of fields are not fields is because $(1, 0)$ must an element of the Cartesian product but since it doesn't have an multiplicative inverse, it fails to satisfy Field Axiom 4b. However, the complex numbers have an equivalent of $(1, 0)$ but its still considered a field. I may be wrong about something here but can anyone explain? Also, doesn't the Field Axiom fail for $(0, 1)$ as well? Or is it that the x part can't equal to zero so only $(1,0)$ is something that should be tested?
I might have others but I don't really know enough to know what I don't know yet. I apologize if these questions are very trivial but I can't quite wrap my head around it and I looked at other sources but no such luck.