# division algorithm

Let R={a+b(sqrt(2)) | a, bâ?? Z}.

(a): Show that R is a ring.

(b): Let M = {a+b(sqrt(2)) â?? R | 5|a and 5|b}. Show that M is a maximal ideal of R.

https://brainmass.com/math/basic-algebra/modern-algebra-ring-357108

#### Solution Preview

Let

(a) Show that R is a ring.

(b) Let Show that M is a maximal ideal of R.

Proof:

(a) To show that R is a ring, we'll show that R is a subring of the field R of real numbers.

Since it follows that

Let and where Then

where and

So

Finally,

where are integers.

So

Therefore, R is a subring of the field R of real numbers.

(b) Let To show that M is an ideal, let and Then we have

where are integers such that and

But then we have and

So and ...

#### Solution Summary

A division algorithm is applied.

$2.19