Show that 1 is a gcd of 2 and 1 sqrt5 in Zsqrt5 but 1 canno

Show that 1 is a gcd of 2 and 1 + sqrt(-5) in Z[sqrt(-5)], but 1 cannot be written in the form 2a + (1 + sqrt(-5))b with a, b in Z[sqrt(-5)].

Solution

Let Z[sqrt(-5)] = { a + bsqrt(-5), a, b belongs to Z}

The norm of 2 is 4 and norm of 1 + sqrt(-5) is 6.

The norm does not have any other gcd in Z[sqrt(-5)] other than 1.

Let if possible 1 can be written as

1 = 2a + (1 + sqrt(-5))b, with a, b in Z[sqrt(-5)].

That is (2a + b) +b sqrt(-5) = 1 in Z[sqrt(-5)].

But this can notbe possible.

Therefore, our assumption was wrong.

Hence 1 cannot be written in the form 2a + (1 + sqrt(-5))b with a, b in Z[sqrt(-5)].