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Mathematics
Linear and Abstract Algebra
Automorphisms of Field Extensions .... Lovett, Example 11.1.8 .... ....
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[QUOTE="Math Amateur, post: 6758130, member: 203675"] 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: [ATTACH=CONFIG]6657[/ATTACH] My questions regarding the above example from Lovett are as follows: [U][I][B]Question 1[/B][/I][/U]In the above text from Lovett we read the following:" ... ... The minimal polynomial of [MATH]\alpha = \sqrt{2} + \sqrt{3}[/MATH] is [MATH]m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1[/MATH] and the four roots of this polynomial are [MATH]\alpha_1 = \sqrt{2} + \sqrt{3}, \ \ \alpha_2 = \sqrt{2} - \sqrt{3}, \ \ \alpha_3 = - \sqrt{2} + \sqrt{3}, \ \ \alpha_4 = - \sqrt{2} - \sqrt{3} [/MATH] ... ... ... ... " Can someone please explain why, exactly, these are roots of the minimum polynomial [MATH]m_{ \alpha , \mathbb{Q} } (x) = x^4 - 10x^2 + 1[/MATH] ... ... and further, how we would go about methodically determining these roots to begin with ... ... [U][I][B]Question 2[/B][/I][/U]In the above text from Lovett we read the following:" ... ... Let [MATH]\sigma \in \text{Aut}(F/ \mathbb{Q} )[/MATH]. Then according to Proposition 11.1.4, [MATH]\sigma[/MATH] must permute the roots of [MATH]m_{ \alpha , \mathbb{Q} } (x)[/MATH] ... ... "Can someone explain what this means ... how exactly does [MATH]\sigma [/MATH] permute the roots of [MATH]m_{ \alpha , \mathbb{Q} } (x)[/MATH] ... ... 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: [ATTACH=CONFIG]6658[/ATTACH] [U][I][B]Question 3[/B][/I][/U]In the above text from Lovett we read the following:" ... ... In Example 7.2.7 we observed that [MATH]\sqrt{2}, \sqrt{3} \in F[/MATH] so all the roots of [MATH]m_{ \alpha , \mathbb{Q} } (x)[/MATH] are in [MATH]F[/MATH] ... ... "Can someone please explain in simple terms exactly why and how we know that [MATH]\sqrt{2}, \sqrt{3} \in F[/MATH] ... ... ? NOTE: Lovett mentions Example 7.2.7 so I am providing the text of this example ... as follows: [ATTACH=CONFIG]6659[/ATTACH] [ATTACH=CONFIG]6660[/ATTACH] I hope that someone can help with the above three questions ... Any help will be much appreciated ... ... Peter [/QUOTE]
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Automorphisms of Field Extensions .... Lovett, Example 11.1.8 .... ....
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