Bezout’s Identity for the Gaussian integers

Prove that if \alpha,\beta \in \mathbb{Z}[i] are relatively prime, then there exist \mu, \eta \in \mathbb{Z}[i] such that \alpha\mu + \beta\eta = 1.


We have seen that \mathbb{Z}[i] is a Euclidean domain, and thus a principal ideal domain. If 1 is a greatest common divisor of \alpha and \beta, then (\alpha,\beta) = \mathbb{Z}[i]. The conclusion follows.

Advertisements
Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: