Question
Show that 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 |
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 as follows: \[ 20 = (2 + \sqrt{-19})(2 - \sqrt{-19}) ________ ____ ____ __________ ________ ________ _______ __________ ___ ________.
_______ ______ _________ ________ _______ _______ __________.
__________ ________ ________ ______ __________ ______ ___ _________ __________ _____.
_________ __________ ________ ______ _____ _________ _____ ______ ___.
______ _____ ________ ______ ____ _______ _____.
________ _______ ____ _________ ___ _______ ___ _______ _____ _______ _______.
_____ __________ _______ ___ ____ _____ __________.
____ ______ _________ _______ ______.
_________ __________ _____ ______ _________ __________ ____ ______ _______ _________ ___.
____ _______ _____ ___ _______ ____ __________ __________ __________ ______.
___ _______ ______ __________ _______ ________ ________ ____ _________ _______ ________ _________.
_______ _____ ___ _____ ________ _______ ____ _____ _______.
______ ______ _________ _______ _____ ______ _________ ________ _______ _______ _______.
____ ______ ______ __________ ________ _________ _________.
____ _______ _________ ___ _______ _______ ___.
_____ __________ _________ ____ _________ ___ _________ __________.
______ ________ ________ ________ _______ _____ ___ ____ ______.
Get Full Answer on WhatsApp
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 as follows: \[ 20 = (2 + \sqrt{-19})(2 - \sqrt{-19}) ________ ____ ____ __________ ________ ________ _______ __________ ___ ________.
_______ ______ _________ ________ _______ _______ __________.
__________ ________ ________ ______ __________ ______ ___ _________ __________ _____.
_________ __________ ________ ______ _____ _________ _____ ______ ___.
______ _____ ________ ______ ____ _______ _____.
________ _______ ____ _________ ___ _______ ___ _______ _____ _______ _______.
_____ __________ _______ ___ ____ _____ __________.
____ ______ _________ _______ ______.
_________ __________ _____ ______ _________ __________ ____ ______ _______ _________ ___.
____ _______ _____ ___ _______ ____ __________ __________ __________ ______.
___ _______ ______ __________ _______ ________ ________ ____ _________ _______ ________ _________.
_______ _____ ___ _____ ________ _______ ____ _____ _______.
______ ______ _________ _______ _____ ______ _________ ________ _______ _______ _______.
____ ______ ______ __________ ________ _________ _________.
____ _______ _________ ___ _______ _______ ___.
_____ __________ _________ ____ _________ ___ _________ __________.
______ ________ ________ ________ _______ _____ ___ ____ ______.
Get Full Answer on WhatsApp
Related Questions
Suppose n is even, n = 2km, where 2 + m, k ≥ 2. Let N = (xm) and H = (y). Show that H N is a subgroup of Dn. What is its order?
Let ? (?) be a finite extension F of odd degree(greater than 1). Show that ? (?2) = ? (?).
Solve the following set of congruences:
Find the elementary divisors and invariant factors of
By looking at the factorisation of x9 - x ∈ [x] guess the number of irreducible polynomials of degree 2 over
. Find all the irreducible polynomials of degree 2 over
.
If char(F) ≠ 2, show that a polynomial ax2 + bx + c is irreducible iff where
is the group of squares in
.
Compute the following:
(i) (ii)
IGNOU NEWS
Assignment Submission Last Date Extended Till 30 June 2026 Click Here★★★IGNOU June 2026 TEE Date Sheet Released Click Here★★★