Originally Posted by

**Bernhard** I am working through Gallian's book "Contemporary Abstract Algebra" (Fifth Edition) and I am on Chapter 20 - Extension Fields.

I need help with Example 1 on page 345 - see attached pdf. [My text below is on the pdf - I am trying to understand the equations]

Example 1 states:

Let $\displaystyle f(x) = x^2 + 1 \in Q[x] $

Then in $\displaystyle E = Q[x]/< x^2 + 1>$ we have

$\displaystyle f(x + <x^2 +1> ) = (x + <x^2 +1>)^2 +1$

$\displaystyle = x^2 + <x^2 +1> + 1$

Can someone please help me understand how the above two lines are equal?

Could you also explain how the last two eqations of the 4 are equal - see attached sheet.

Bernhard