Automorphisms of Field Extensions .... Lovett, Example 11.1.8 .... ....

In summary, Peter is reading "Abstract Algebra: Structures and Applications" by Stephen Lovett and has questions about the example on page 559 and the Proposition 11.1.4. Additionally, Peter is looking for clarification on how to find the roots of a quadratic equation.
  • #1
Math Amateur
Gold Member
MHB
3,998
48
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:
Question 1In 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 2In 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 3In 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
 
Last edited:
Physics news on Phys.org
  • #2
Hi Peter,

Peter said:
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 ... ...

Fortunately, you can use high school algebra. The polynomial $x^4 - 10x^2 + 1$ is a quadratic in disguise, namely, if $u = x^2$ then the polynomial is $u^2 - 10u + 1$. So you can use the quadratic formula to find the roots.
Peter said:
Question 2In 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 ... ...

Let $R$ be the set of roots of $m_{\alpha, \Bbb Q}$ in $F$. Since $\sigma$ is an automorphism of $F$, it is a bijection. By Proposition 11.6, $\sigma$ restricted to $R$ is a map $R\to R$, which is a bijection since $\sigma$ is a bijection. Hence, $\sigma$ permutes the elements of $R$, i.e., the roots of $m_{\alpha, \Bbb Q}$.
Peter said:
Question 3In 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:

It follows from closure under addition and scalar multiplication in $F$. For example, since $\alpha_1, \alpha_2\in F$, then $(1/2)(\alpha_1 + \alpha_2)\in F$, i.e., $\sqrt{2}\in F$.
 
  • #3
Euge said:
Hi Peter,
Fortunately, you can use high school algebra. The polynomial $x^4 - 10x^2 + 1$ is a quadratic in disguise, namely, if $u = x^2$ then the polynomial is $u^2 - 10u + 1$. So you can use the quadratic formula to find the roots.
Let $R$ be the set of roots of $m_{\alpha, \Bbb Q}$ in $F$. Since $\sigma$ is an automorphism of $F$, it is a bijection. By Proposition 11.6, $\sigma$ restricted to $R$ is a map $R\to R$, which is a bijection since $\sigma$ is a bijection. Hence, $\sigma$ permutes the elements of $R$, i.e., the roots of $m_{\alpha, \Bbb Q}$.

It follows from closure under addition and scalar multiplication in $F$. For example, since $\alpha_1, \alpha_2\in F$, then $(1/2)(\alpha_1 + \alpha_2)\in F$, i.e., $\sqrt{2}\in F$.
Thanks Euge ... I appreciate your help ...

Peter
 

Related to Automorphisms of Field Extensions .... Lovett, Example 11.1.8 .... ....

1. What is an automorphism in the context of field extensions?

An automorphism in the context of field extensions is a bijective map from a field to itself that preserves the field operations of addition and multiplication. In other words, it is a way of rearranging the elements of a field while maintaining the same algebraic structure.

2. Can you give an example of an automorphism of a field extension?

Yes, Example 11.1.8 in Lovett's book provides an example of an automorphism of a field extension. It demonstrates how the field of real numbers can be extended to the field of complex numbers, and how the automorphism maps the real numbers to a subset of the complex numbers.

3. How are automorphisms of field extensions useful?

Automorphisms of field extensions are useful in various areas of mathematics, including algebra, number theory, and geometry. They can be used to study the structure of fields and to prove certain properties of fields. They also have applications in cryptography and coding theory.

4. Are all field extensions guaranteed to have automorphisms?

No, not all field extensions have automorphisms. For a field extension to have an automorphism, it must have certain properties, such as being algebraic and having a finite degree over the base field. There are also some field extensions, known as non-Galois extensions, that do not have any automorphisms.

5. How do automorphisms of field extensions relate to Galois theory?

Automorphisms of field extensions are closely related to Galois theory, which is a branch of abstract algebra that studies the symmetry of field extensions. In Galois theory, automorphisms are used to study the structure of fields and to solve equations by finding their symmetries. They also play a crucial role in the fundamental theorem of Galois theory, which relates the roots of a polynomial to its automorphisms.

Similar threads

  • Linear and Abstract Algebra
Replies
5
Views
2K
  • Linear and Abstract Algebra
Replies
4
Views
2K
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
1
Views
891
  • Linear and Abstract Algebra
Replies
6
Views
2K
  • Linear and Abstract Algebra
Replies
2
Views
1K
  • Linear and Abstract Algebra
Replies
15
Views
2K
  • Linear and Abstract Algebra
Replies
3
Views
2K
  • Linear and Abstract Algebra
Replies
1
Views
1K
  • Linear and Abstract Algebra
Replies
4
Views
1K
Back
Top