division algorithm
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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.
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Solution Summary
A division algorithm is applied.
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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 ...
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