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I am reading "Abstract Algebra: Structures and Applications" by Stephen Lovett ...

I am currently focused on Chapter 8: Galois Theory, Section 1: Automorphisms of Field Extensions ... ...

I need help with Example 11.1.8 on page 559 ... ...

Example 11.1.8 reads as follows:

View attachment 6657

My questions regarding the above example from Lovett are as follows:

In the above text from Lovett we read the following:

" ... ... The minimal polynomial of \(\displaystyle \alpha = \sqrt{2} + \sqrt{3}\) is \(\displaystyle m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1\) and the four roots of this polynomial are

\(\displaystyle \alpha_1 = \sqrt{2} + \sqrt{3}, \ \ \alpha_2 = \sqrt{2} - \sqrt{3}, \ \ \alpha_3 = - \sqrt{2} + \sqrt{3}, \ \ \alpha_4 = - \sqrt{2} - \sqrt{3} \)

... ... ... ... "

Can someone please explain why, exactly, these are roots of the minimum polynomial \(\displaystyle m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1\) ... ... and further, how we would go about methodically determining these roots to begin with ... ...

In the above text from Lovett we read the following:

" ... ... Let \(\displaystyle \sigma \in \text{Aut}(F/ \mathbb{Q} )\). Then according to Proposition 11.1.4, \(\displaystyle \sigma\) must permute the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) ... ... "

Can someone explain what this means ... how exactly does \(\displaystyle \sigma \) permute the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) ... ... and how does Proposition 11.1.4 assure this, exactly ... ...

NOTE: The above question refers to Proposition 11.1.4 so I am providing that proposition and its proof ... ... as follows:

View attachment 6658

In the above text from Lovett we read the following:

" ... ... In Example 7.2.7 we observed that \(\displaystyle \sqrt{2}, \sqrt{3} \in F\) so all the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) are in \(\displaystyle F\) ... ... "

Can someone please explain in simple terms exactly why and how we know that \(\displaystyle \sqrt{2}, \sqrt{3} \in F\) ... ... ?

NOTE: Lovett mentions Example 7.2.7 so I am providing the text of this example ... as follows:

View attachment 6659

View attachment 6660

I hope that someone can help with the above three questions ...

Any help will be much appreciated ... ...

Peter

I am currently focused on Chapter 8: Galois Theory, Section 1: Automorphisms of Field Extensions ... ...

I need help with Example 11.1.8 on page 559 ... ...

Example 11.1.8 reads as follows:

View attachment 6657

My questions regarding the above example from Lovett are as follows:

**Question 1**In the above text from Lovett we read the following:

" ... ... The minimal polynomial of \(\displaystyle \alpha = \sqrt{2} + \sqrt{3}\) is \(\displaystyle m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1\) and the four roots of this polynomial are

\(\displaystyle \alpha_1 = \sqrt{2} + \sqrt{3}, \ \ \alpha_2 = \sqrt{2} - \sqrt{3}, \ \ \alpha_3 = - \sqrt{2} + \sqrt{3}, \ \ \alpha_4 = - \sqrt{2} - \sqrt{3} \)

... ... ... ... "

Can someone please explain why, exactly, these are roots of the minimum polynomial \(\displaystyle m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1\) ... ... and further, how we would go about methodically determining these roots to begin with ... ...

**Question 2**In the above text from Lovett we read the following:

" ... ... Let \(\displaystyle \sigma \in \text{Aut}(F/ \mathbb{Q} )\). Then according to Proposition 11.1.4, \(\displaystyle \sigma\) must permute the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) ... ... "

Can someone explain what this means ... how exactly does \(\displaystyle \sigma \) permute the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) ... ... and how does Proposition 11.1.4 assure this, exactly ... ...

NOTE: The above question refers to Proposition 11.1.4 so I am providing that proposition and its proof ... ... as follows:

View attachment 6658

**Question 3**In the above text from Lovett we read the following:

" ... ... In Example 7.2.7 we observed that \(\displaystyle \sqrt{2}, \sqrt{3} \in F\) so all the roots of \(\displaystyle m_{ \alpha , \mathbb{Q} } (x)\) are in \(\displaystyle F\) ... ... "

Can someone please explain in simple terms exactly why and how we know that \(\displaystyle \sqrt{2}, \sqrt{3} \in F\) ... ... ?

NOTE: Lovett mentions Example 7.2.7 so I am providing the text of this example ... as follows:

View attachment 6659

View attachment 6660

I hope that someone can help with the above three questions ...

Any help will be much appreciated ... ...

Peter

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