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I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ...
I am currently focused on Chapter 7: Field Extensions ... ...
I need help with Example 7.1.5 ...Example 7.1.5 reads as follows:
https://www.physicsforums.com/attachments/6572
https://www.physicsforums.com/attachments/6573
In the above text from Lovett, we read the following:
" ... ... Then $$\mathbb{Q} [x] / ( p(x) ) = \mathbb{Q} [ \sqrt{5} ] $$ is a field. ... ... "
I understand that $$\mathbb{Q} [x] / ( p(x) ) = \mathbb{Q} [x] / ( x^2 - 5 ) $$ is a field ... ... but why is it equal to $$\mathbb{Q} [ \sqrt{5} ]$$ ... ...?Can someone please explain and demonstrate why the equality $$\mathbb{Q} [x] / ( x^2 - 5 ) = \mathbb{Q} [ \sqrt{5} ]$$ holds ... ?Help will be appreciated ...
Peter
I am currently focused on Chapter 7: Field Extensions ... ...
I need help with Example 7.1.5 ...Example 7.1.5 reads as follows:
https://www.physicsforums.com/attachments/6572
https://www.physicsforums.com/attachments/6573
In the above text from Lovett, we read the following:
" ... ... Then $$\mathbb{Q} [x] / ( p(x) ) = \mathbb{Q} [ \sqrt{5} ] $$ is a field. ... ... "
I understand that $$\mathbb{Q} [x] / ( p(x) ) = \mathbb{Q} [x] / ( x^2 - 5 ) $$ is a field ... ... but why is it equal to $$\mathbb{Q} [ \sqrt{5} ]$$ ... ...?Can someone please explain and demonstrate why the equality $$\mathbb{Q} [x] / ( x^2 - 5 ) = \mathbb{Q} [ \sqrt{5} ]$$ holds ... ?Help will be appreciated ...
Peter
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