Example 25.12 described an ordering of Z[√2] = {m + n√2 |m, n ∈ Z} in which -√2 is positive. Describe, in terms of m and n, all positive elements of Z[√2] in that ordering.

Data from Example 25.12

Exercise 11 of Section 18 shows that {m + n√2| m, n ∈ Z} is a ring. Let us denote this ring by Z[√2]. This ring has a natural order induced from R. in which √2 is positive. However, we claim that ∅: Z[√2] → Z[√2] defined by ∅(m + n√2) = m - n√2 is an automorphism. It is clearly one to one and onto Z[√2]. We leave the verification of the homomorphism property to Exercise 17. Because ∅(√2) = -√2, we see the ordering induced by ∅ will be one where -√2 is positive! In the natural order on Z[√2], an element m + n√2 is positive if m and n are both positive, or if m is positive and 2n² < m² , or if n is positive and m² < 2n². In Exercise 3, we ask you to give the analogous descriptions for positive elements in the ordering of Z[√2] induced by ∅.