Question

Let `p` be a prime.Show that `S={m+nsqrt(-p) | m,n in ZZ}` is a subring of `CC`..Further,check whether or not `S` is an ideal of `CC`?

Answer 1

`S` is a subring but not an ideal.

Explanation:

Given:

`S = { m + nsqrt(-p) | m, n in ZZ }`

• `S` contains the additive identity:
  `0 + 0sqrt(-p) = 0color(white)(((1/1),(1/1)))`

• `S` is closed under addition:
  `(m_1 + n_1 sqrt(-p)) + (m_2 + n_2 sqrt(-p)) = (m_1+m_2)+(n_1+n_2) sqrt(-p)color(white)(((1/1),(1/1)))`

• `S` is closed under additive inverse:
  `(m_1 + n_1 sqrt(-p)) + (-m_1 + -n_1 sqrt(-p)) = 0color(white)(((1/1),(1/1)))`

• `S` is closed under multiplication:
  `(m_1 + n_1 sqrt(-p))(m_2 + n_2 sqrt(-p)) = (m_1m_2-pn_1n_2)+(m_1n_2+m_2n_1)sqrt(-p)color(white)(((1/1),(1/1)))`

So `S` is a subring of `CC`.

It is not an ideal, since it does not have the property of absorption.

For example:

`sqrt(3)(1+0sqrt(-p)) = sqrt(3) !in S`