IGNOU BMTC 134 2026 - Find all the units of . Check whether or n

Question

Find all the units of $\mathbb{Z}[\sqrt{-7}]$ . $\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad$ Check whether or not $\mathbb{Q}[x] / \langle 8x^3 + 6x^2 - 9x + 24 \rangle$ is a field. $\quad\quad\quad\quad\quad\quad$

10 Jan 2026

Answer :

Word Count : 418

The ring (\mathbb{Z}[\sqrt{-7}]) consists of elements of the form (a + b\sqrt{-7}), where (a, b \in \mathbb{Z}). An element (u = a + b\sqrt{-7}) is a unit if it has a multiplicative inverse in the ring. This is equivalent to saying there exists (v = c + d\sqrt{-7} \in \mathbb{Z}[\sqrt{-7}]) such that (uv = 1). Compute the norm (N(a + b\sqrt{-7}) = a^2 + 7b^2). The key property: if (uv = 1), then (N(u)N(v) = N(uv) = N(1) = 1). So the norm must satisfy (N(a + b\sqrt{-7}) = a^2 + 7b^2 = 1). Now solve (a^2 + 7b^2 = 1) in integers: * If (b = 0), then (a^2 = 1 \implies a = \pm 1). * If (b \neq 0), then (7b^2 \ge 7 > 1), impossible. Thus the only solutions are (a = \pm 1, b = 0). Units in (\mathbb{Z}[\sqrt{-7}]): [ \boxed{\pm 1}. ] --- Next, consider (\mathbb{Q}[x]/\langle 8x^3 + 6x^2 - 9x + 24 \rangle). A quotient ring (\mathbb{Q}[x]/\langle f(x) \rangle) is

____ ___ ____ ___ ________.
______ ________ ____ _____ ______ _______ _____ ___ _____ ____.
______ ______ _______ ___ _____ ______ _______ _____ _________ ____ _______.
_____ _______ _______ _________ ____ __________ _______ _________ ___.
____ ____ _______ _______ ______ _______ ____ _____.
______ _______ ____ ______ ____ ___ ______ ____.
________ __________ _________ _____ ______ _____ _________ ________ ___ ___ ___ _________.
_________ ______ ____ ___ ______ _______.
________ ___ ___ _____ ______ ______ _______ ____ ________.
_____ _____ ______ _____ ____ _________ _________ _____ _________ _______.
_____ __________ _____ _________ _________ ____ ______ __________ ______.
___ _________ ________ _____ ________ _________ ____ ____ _____.
___ _____ _________ _____ __________.
________ _________ ____ ________ _________ _________ ________ _____ __________ ____.
__________ _______ _______ _____ ________ __________ __________ ________ ________.
___ _____ _______ __________ __________ __________ ________ _________ _____ ________ ________.
________ ________ _________ ________ ______.
_______ ________ _____ ____ _______ _____ _________ ________ _____ ____ ___.
_________ __________ ________ _____ ____ ______ __________ ___ _______ ___ ____ _______.
__________ ____ _________ ____ ________ ___ ______ ________.
_________ _______ __________ _______ __________.
________ ___ ___ ___ ____ _____ ______ ____ _____.
____ ___ _______ ______ ______ ____ ___ ___ _________.
_______ ___ _________ ___ _______ ______ __________ _______ _________ ________ ______ _________.
_____ __________ _____ ______ ___ ___ ____.
______ _______ ___ ____ ___ __________.
_________ __________ _____ ___ ________ ____ __________ ______.
_______ ________ ______ _________ __________ ___ ____ ___ _____ ________ ___.
________ __________ _____ ________ _________ ___ __________.

IGNOU ACADEMY

Your IGNOU journey, made easier.

Solved Assignments
Study Notes
Guess Papers
Question Papers

Call Us WhatsApp

STUDY • PREPARE • SUCCEED

Skip in 10