I am reading Dummit and Foote Chapter 8, Section 8.3 UFDs.
On page 284 (see attached) Dummit and Foote prove Proposition 10 which shows the following:
"In an integral domain a prime element is always irreducible."
D&F then state:
"It is not true in general that an irreducible element is necessarily prime."
They give as an example the element 3 in the quadratic integer ring
They assert that the element 3 is irreducible but not prime.
I am struggling to show rigorously that 3 is irreducible in R, despite D&F's reference to the calculations on page 273 (see attached)
Can someone please help me produce a formal and rigorous demonstration that 3 is irreducible.
That is to show that whenever
then one of must be a unit in
Thanks for pointing that out ... And apologies for forgetting this ... My day job intrudes on my Maths to the point I sometimes have to revisit problems.
Will now go to the posts you mention
if 3 were reducible in Z[√(-5)], say 3 = ab where a,b are not units, we would have N(3) = 9 = N(ab) = N(a)N(b) (here N is the field norm, or the square of the complex modulus. this is always a positive integer).
if N(a) OR N(b) = 1, then it must be ±1, and both of these are units.
hence N(a) = N(b) = 3.
suppose a = c + d√(-5) where c,d are integers.
N(a) = 3 implies c2+5d2 = 3.
thus d = 0, (or else N(a) > 3) and c2 = 3 has no integer solution.