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I am currently focused on Chapter 7: Field Extensions ... ...

I need help with some remarks of Lovett following Theorem 7.1.12 and Example 7.1.13 on page 326 ...

The remarks by Lovett read as follows:

https://www.physicsforums.com/attachments/6589

In the above remarks from Lovett, we read the following:

" ... ... In the quotient ring \(\displaystyle K\), this implies that \(\displaystyle \overline{ a(x) q(x) } = 1\). Thus in \(\displaystyle K, \ a( \alpha ) q( \alpha ) = 1\). ... ... "

My question is as follows:

Can someone please explain exactly why/how \(\displaystyle \overline{ a(x) q(x) } = 1\) implies that \(\displaystyle a( \alpha ) q( \alpha ) = 1\) ... ... ?

Help will be appreciated ...

Peter