The Gauss Lemma extends irreducibility from this polynomials with these as (*) coefficients to polynomials over Q.  Cantor's Diagonalization proof showed this set has the same cardinality as the rationals, aleph null.  
The cyclotomic variety of these objects cannot be uniquely factored if p is greater than 23.  Similar objects named for Eisentstein and Gauss are equal to a plus b times the cube root of minus one, and a plus b times the square root of minus one, respectively.  The Gauss Lemma extends irreducibility from this polynomials with these as (*) coefficients to polynomials over Q.  Cantor's Diagonalization proof showed this set has the same cardinality as the rationals, aleph null.  The fundamental theorem of arithmetic implies that these can always be uniquely factored.  For 10 points, name this countably infinite ring, the set of numbers that contain no fractional or decimal part forms, symbolized Z.
The cyclotomic variety of these objects cannot be uniquely factored if p is greater than 23.  
Integers [accept Z until mentioned, do not accept 'Natural Numbers," "Rationals," "Reals," "N," "Q," or "R"]