1. ## Field Axioms

The Question: Show the rational numbers satisfy the field axioms. You may assume the commutative, associative, and distributive laws hold for the integers.

Earlier, the book defined the field axioms for the real numbers. Is it correct to say that the rationals are a subset of the reals, therefore they satisfy the field axioms?

2. Originally Posted by zg12
The Question: Show the rational numbers satisfy the field axioms. You may assume the commutative, associative, and distributive laws hold for the integers.

Earlier, the book defined the field axioms for the real numbers. Is it correct to say that the rationals are a subset of the reals, therefore they satisfy the field axioms?
No, because (for example) the set of integers is also a subset of the reals.

If you use that the reals are a field, then you can skip some checks and just check closure, identity, inverses. The question statement kind of makes it sound like they want you to do it the long way, though, just using that the set of integers is a ring.

3. Yes, the long way as this is a rigorous approach to very basic high school math. So you would just go through each axiom and replace each term by a ratio? For example instead of the commutative axiom for addition x+y=y+x replace it with x/y +w/z= w/z + x/y or x/y +0=x/y, etc?

4. Originally Posted by zg12
Yes, the long way as this is a rigorous approach to very basic high school math. So you would just go through each axiom and replace each term by a ratio? For example instead of the commutative axiom for addition x+y=y+x replace it with x/y +w/z= w/z + x/y or x/y +0=x/y, etc?
Yes, and be sure to use the (formal) definitions of addition/multiplication for rationals. If you get stuck you can ask a question here, or check out this blog post (not mine)

Math Refresher: The Set of Rational Numbers

which does everything step by step. (Of course don't use it to cheat, etc.)