Answer to Question #105101 in Abstract Algebra for Sourav Mondal

Question

Check whether or not S={a0+a1x+.......+anx^n belongs to Z[x]|5|a0}is an ideal of Z[x]

Answer 1

$S$ $=$ { $a_0+a_1x+.......+a_nx^n$ $\in Z[x]: 5|a_0$ }

According to ideal test ,

Let $f(x)=a_0+a_1x+.....+a_nx^n,g(x)=b_0+b_1x+.........+b_mx^m$

are two elements of S and $r(x) = c_0+c_1x+........+c_lx^l\in Z[x]$

Then $f(x)-g(x)\in S$ ,Since $5|a_0 \space and \space 5|b_0\implies 5|a_0-b_0$

and $r(x)f(x)\in S$ , Since $5|a_0\implies 5|a_0c_0$

Similarly,$f(x)r(x)\in S.$

Hence ,S is an ideal of $Z[x]$