Question

a) Find all the units of Z[ − 7]. b) Check whether or not < 8x + 6x − 9x + 24 > [x] 3 2 Q is a field. c) Construct a field with 125 element

Answer 1

Solution:-

(a)

If $a\in R$ be a unit and a, b $\in$ R , such that

Z(-7) = {a+(-7)b} , a,b $\in$ Z

let (a+(-7)) $\in$ Z(-7) be a unit $\exist$ (c+(-7)d) $\in$ (-7)

(a+(-7)b)(c+(-7)d)=1.............(1)

$\implies$

(a-7b)(c-7d)=1

$\implies$

a=1 and c can be 1

as b,d $\in$ Z so cannot be fractional

(b)

8x + 6x − 9x + 24 > [x] 3 2 Q is a field

[x] 3 2 Q is a field

on solving we get

x³-2

5x-24 < x³-2

$\implies$

x³-5x+22>0

as we can see

$\boxed{X= -3.3 , 1.6+1.9i , 1.6-1.9i}$

(c)

$Solution: ~We~ know ~ that \\To ~find ~ an ~ irreducible ~ polynomial ~ of ~ degree ~3 ~ in ~Z_5[x]. \\x^3 +x+1~ is ~ one ~ such ~polynomial~, it~ clearly ~has ~ no ~ linear~ factors~(Since ~0,1,2,3,4,) ~are ~\\not ~roots. \\So, F=\frac{Z_5[x]}{x^3 +x+1} ~is ~ a~ field ~ with ~ 5^3 ~elements.$