Show that is not a UFD by giving two different factorisations of 20.

Question

Show that $\mathbb{Q}(\sqrt{-19})$ is not a UFD by giving two different factorisations of 20.

Question no.	Block 1	Block 2	Block 3	Block 4	Block 5
2 a)	5
2 b)	3
2 c)		2
3 c)			10
4 a)	4
4 b)	3
4 c)			3
5 a)				2
5 b)				5
5 c)				3
6 a)				2
6 b)				6
6 c)				2
7 a)		4
7 c)		1
7 c)		3
7 c)		2
8 a)	3
8 b)	4
8 c)	2
8 d)	3
8 e)	3
9 a)		5
9 b)			5
9 c)			5
Total	30	17	23	20	0

20 Feb 2025

Answer :

Word Count : 199

To show that $ \mathbb{Q}(\sqrt{-19}) $ is not a Unique Factorization Domain (UFD), we need to find two distinct factorizations of 20 in this number field. ### Factorization 1: In the ring $ \mathbb{Q}(\sqrt{-19}) $, we can factor 20 _________ _____ __________ _____ ___ _________ ______ ____ ____ ______.
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