For the greatest... teachers, pupils and "buddys"...Gauss,Riemann,...

Algebra Level 4

Let $\mathbb{Z} [i] = \{a + bi; a,b \in \mathbb{Z}\text{ and } i =\sqrt{-1}\}$ to be integers Gaussian set. Which statement(s) below is/are true?

a) $\mathbb{Z}[i]$ is an Euclidean Domain.

b) $\mathbb{Z} [i]$ is an Principal Ideal Domain.

c) $\mathbb{Z} [i]$ is an Unique Factorization Domain

All of them None of them b) c) a)

1 solution

Akash Patalwanshi
May 28, 2016

We know that Gaussian integers forms an Euclidean domain.

And we know, $ED⇒ PID⇒ UFD$ So, all statements are true.

yes, that is the answer summarily, thank you!(+1)

Guillermo Templado - 5 years ago

Sir, I love Abstract algebra :-) Sorry that, I had not provided proof of Gaussian integers form an $ED$ but it can found easily in any standard text.

akash patalwanshi - 5 years ago

don't worry,yes, the proof can be found in a text book... Don't call me sir, call me Guillermo, please. I'm glad to we are buddies now $\ddot \smile$ . And I love abstract algebra too, a lot...

Guillermo Templado - 5 years ago

@Guillermo Templado – Thank you so much. :-)

akash patalwanshi - 5 years ago

For the greatest... teachers, pupils and "buddys"...Gauss,Riemann,...

1 solution

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