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