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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.

Thank you!

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    $\begingroup$ The cartesian product of two fields is not a field, since $(1,0)\cdot (0,1) = (0,0)$. $\Bbb C$ is a field, but not arising as the product $\Bbb R\times\Bbb R$. Subtle difference. $\endgroup$ – Ted Shifrin Sep 22 '17 at 1:42
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$(1,0)$ isn't a multiplicative identity because it doesn't work. For instance, $(1,0) \cdot (0,1) = (0,0)$, not $(0,1)$.

Direct products of fields are never fields (if you require $0 \neq 1$ for a field, which most sensible people do.) The reasons are plentiful: zero divisors will exist and not everyone is invertible.

Your confusion might lie here: $\mathbb{C}$ may be thought of as $\mathbb{R} \times \mathbb{R}$ either as a vector space over $\mathbb{R}$ or as a topological space. In these senses it IS a cross product, but it is NOT so as a ring.

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