An integral domain is a commutative ring with no zero-divisors:or
The ringis an integral domain. (This explains the name.)
The polynomial ringsandare integral domains.
(Look at the degree of a polynomial to see how to prove this.)
The ringis an integral domain.
Ifis prime, the ringis an integral domain.
1 and 3 follow from the properties of real numbers.
2 follows from the linear independence of distinct powers of IfandthenwhereIfthen the coefficient offor eachso we havelinearly independent equations each equal 0 andunknowns (if we treat theas unknown and solve them in terms of the) orunknown (if we treat theas unknown and solve them in terms of theIn either case there are more equations than unknowns so each unknown coefficient is equal to zero.
4 follows from the fact thatorby Euclid's Lemma henceor
Ifare elements of a field withthen ifit has an inverseand multiplying both sides by this givesHence there are no zero-divisors and we have that every field is an integral domain.
Ifis an integral domain then cancellation for multiplication is permissible. IfandthenThis follows from thatexists, so