Difficult)
Theorem 3.1 along with the example preceding it are a new concept to me. The context of a field or a ring made out of a cartesian product is confusing to me and I need a lot more explanation for it to make sense...
Are we doing the cross product? FROM WIKIPEDIA I copied the definition. Would have been nice in the book.
That is, for sets A and B, the Cartesian product A × B is the set of all ordered pairs (a, b) where a ∈ A and b ∈ B.
Interesting or question about the reading)
I am curious. The reals are a ring, and also a field, and the quotient numbers are a subring of the reals, and also a subfield. Where the integers are a subring of the quotient numbers, proving a subring is actually easier than proving the superset is a ring. We only need to show closure under addition and multiplication, the additive identity, and the additive inverse.
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