Theorem·T27
The rationals form a field
Rational addition and multiplication satisfy all the field axioms: a commutative ring with unity where every nonzero element has an inverse.
(+, ·) is a field: it is a commutative ring with unity
(
from D064), and every nonzero element has a multiplicative inverse via D069.
In words
Rational addition and multiplication satisfy all the field axioms: a commutative ring with unity where 1 and 0 are different, and every nonzero element has an inverse.
Never needed: F10 · A03 · A04 · A05 · A07 (computed from the citation graph, not asserted).
Proof
- 1is an abelian group by T26.
- 2Associativity and commutativity of . Let . By D067 twice, and , the same representative pair by associativity of on . Commutativity: and are the same pair by commutativity.
- 3Distributivity. (D065), so by D067: Also , , so by D065: These pairs are -equivalent, not literally equal (the second uses the redundant denominator in place of ): writing for the first pair and for the second, both and expand, via distributivity, commutativity and associativity of , to the same sum so , i.e. (L55), giving by T04. The other distributive law follows symmetrically (or from this one via commutativity).
- 4Unity. , using the unity law of T23; by commutativity, too.
- 5
- 6Inverses. For (so ), let (D069). By D067, . Since by commutativity, both coordinates of this pair are the same integer , so , and exactly when , which holds by the unity law and commutativity of T23. So .
- 7All clauses of D068 hold: is a field.
∎
Remarks
The first field in this library, and the number system needed for coordinate geometry: the arithmetic backing for the Pythagorean theorem (squared distances, dot products) can now be built over
without needing square roots or a construction of
.