is not principal, which can be proved by contradiction as follows. formula_1 has a norm function formula_2, which satisfies formula_3, and formula_4 if and only if formula_5 is a unit in formula_1. First of all, formula_7, because the quotient ring of formula_1 modulo the ideal formula_9 is isomorphic to formula_10, so that the quotient ring of formula_1 modulo formula_12 is isomorphic to formula_13. If "J" were generated by an element "x" of "R", then "x" would divide both 2 and 1 + . Then the norm formula_14 would divide both formula_15 and formula_16, so "N"(x) would divide 2. If formula_17, then formula_18 is a unit, and formula_19, a contradiction. But formula_14 cannot be 2 either, because "R" has no elements of norm 2, because the Diophantine equation formula_21 has no solutions in integers, as it has no solutions modulo 5.