Need help with 514 Solution514 Let px be a polynomial of deg

Need help with 5.14

Solution

5.14

Let, p(x) be a polynomial of degree n in R[x] and let, p be a unit\'

So there is some polynomial g(x) in R[x] of degree, m so that

p(x)g(x)=1

But degree of p(x)g(x)=n+m=0

BEcause , 1 is a polynomial of degree 0

But, n,m>=0

SO, n=m=0

So, p is a non zero constant polynomial

ii)

(2x+1)^2=4x^2+4x+1=1 modulo 4

HEnce,

(2x+1)^2=1 in Z4[x]

Hence, 2x+1 is a unit as 2x+1 is its own inverse