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

Peter