Wikipedia:Reference desk/Archives/Mathematics/2017 May 24

= May 24 =

Irreducibles without primes
Is there an integral domain with an irreducible element but no prime elements? GeoffreyT2000 (talk) 01:26, 24 May 2017 (UTC)
 * How about $$\mathbb{C}[x,y,z]/(y^2-x, z^2-x)$$? y and z are irreducible and are obviously not prime.  I'm fairly certain there are no prime elements: given a non-unit polynomial $$p(y,z) = y^mz^n + q(y,z)$$, $$[y^mz^n + q(y,z)] \cdot [y^mz^n - q(y,z)] = [y^{m-1}z^{n+1} + q(y,z)] \cdot [y^{m-1}z^{n+1} - q(y,z)]$$.  And by degree considerations, if $$p(y,z)$$ divides either of these factors, it differs from them by a unit.  But counting the parity of the number of occurrences of y rules out that possibility.--2406:E006:332E:1:423:6A72:E875:1DD9 (talk) 05:17, 24 May 2017 (UTC)
 * The ring that you give here is not an integral domain because y and z have equal squares but are neither themselves equal nor additive inverses of each other, so y-z and y+z are nonzero elements with a product of zero. (The usual difference times sum formula for a difference of squares works in any commutative ring.) GeoffreyT2000 (talk) 16:31, 24 May 2017 (UTC)
 * Any noetherian domain has irreducibles proof here. If you take K[t^2,t^3] (= K[x,y]/(y^2-x^3), the cuspidal cubic) and localize at the singular point, or formally complete to $$Kt^2,t^3$$, you get a local ring with only 1 prime ideal: (t^2, t^3), which clearly is not principal, clearly has no generator which is prime, while t^2 and t^3 are irreducible. See or  also.John Z (talk) 23:58, 24 May 2017 (UTC)

Square root of -7 in 2-adic numbers
Does the Ramanujan–Nagell equation give rise to a square root of -7 (= ...111001) in $$\mathbb{Q}_2$$, the 2-adic numbers? GeoffreyT2000 (talk) 01:29, 24 May 2017 (UTC)


 * It would if there were an infinite number of solutions, but it doesn't seem that there are. You don't need that though since all you need is
 * $$x^2=-7 \mod 2^n$$
 * not
 * $$x^2=2^n-7$$
 * Two square roots of -7 are
 * ...0 1100 0000 1011 0101 & ...1 0011 1111 0100 1011. --RDBury (talk) 07:26, 25 May 2017 (UTC)