Finite fields

**Mike12** · January 29th 2011, 04:28 PM

can any one help me to know how to do this question please:
if I have theta is a root of the polynomial p(x) ,where p(x)=x^3-2x-2 is irreducible in Q[x] how can I compute (1+theta)(1+theta+theta^2) and (1+theta)/1+theta+theta^2 in Q(theta)

**tonio** · January 30th 2011, 01:54 AM

Originally Posted by Mike12

can any one help me to know how to do this question please:
if I have theta is a root of the polynomial p(x) ,where p(x)=x^3-2x-2 is irreducible in Q[x] how can I compute (1+theta)(1+theta+theta^2) and (1+theta)/1+theta+theta^2 in Q(theta)

$\displaystyle{p(\theta)=\theta^3-2\theta-2=0\Longrightarrow \theta^3=2\theta+2}$ , so for example

$\displaystye{(1+\theta)(1+\theta+\theta^2)=1+2\the ta+2\theta^2+\theta^3=1+2\theta+2\theta^2+2\theta+ 2=3+4\theta+2\theta^2}$

Tonio

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