Is the follwing set a field?
$Q ∪ [-1,1]$
I read the notes in my textbook about fields and understand most of the field axioms. However I want to see an example which is worked. Furthermore I am confused whether a field HAS to HAVE $0$ and $1$ in it?
So for this question I am leaning towards saying it is a field, as I cannot find a pair of numbers that violates the axioms. However would I prove it formally… or am I wrong and is this not a field?
Best Answer
No, it's not a field, because it's not closed under addition. $100$ and $1/\sqrt{2}$ belong to your set, but $100 + 1/\sqrt{2}$ doesn't.
Yes, this set has an additive identity $0$ and a multiplicative identity $1$, because you are adding and multiplying numbers in the usual manner, but that doesn't matter much because the set is not closed under addition.