Algebraic Number Theory by Jurgen Neukirch

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Introduction

In the realm of abstract algebra and Galois theory, the concept of Q-automorphism plays a crucial role in understanding the properties of algebraic numbers. An algebraic number is a complex number that is a root of a non-constant polynomial equation with rational coefficients. The Q-automorphism of algebraic numbers refers to the automorphisms of the algebraic closure of the rational numbers, Q, that fix Q pointwise. In this article, we will delve into the world of Q-automorphisms and explore their properties and applications.

What are Q-automorphisms?

A Q-automorphism is an automorphism of the algebraic closure of Q that fixes Q pointwise. In other words, it is a bijective homomorphism from the algebraic closure of Q to itself that leaves every element of Q unchanged. The algebraic closure of Q is a field that contains all the roots of all non-constant polynomials with rational coefficients. It is denoted by Q̄.

Properties of Q-automorphisms

Q-automorphisms have several important properties that make them a fundamental concept in Galois theory. Some of these properties include:

Fixing Q pointwise: As mentioned earlier, Q-automorphisms fix Q pointwise, meaning that they leave every element of Q unchanged.
Bijective: Q-automorphisms are bijective, meaning that they are both injective and surjective.
Homomorphism: Q-automorphisms are homomorphisms, meaning that they preserve the algebraic structure of the algebraic closure of Q.
Automorphism: Q-automorphisms are automorphisms, meaning that they are bijective homomorphisms.

Examples of Q-automorphisms

There are several examples of Q-automorphisms that can be used to illustrate their properties. Some of these examples include:

Identity automorphism: The identity automorphism is a Q-automorphism that maps every element of the algebraic closure of Q to itself.
Complex conjugation: Complex conjugation is a Q-automorphism that maps every complex number to its complex conjugate.
Field automorphisms: Field automorphisms are Q-automorphisms that map every element of the algebraic closure of Q to a different element of the same field.

Exercise: Proving the existence of Q-automorphisms

Let E be an algebraic extension of K, and let F be an algebraic closure of K containing E. Prove that if σ: E → F is an embedding such that σ(c) = c for all c in K, then σ is a Q-automorphism.

Solution

To prove that σ is a Q-automorphism, we need to show that it is a bijective homomorphism from the algebraic closure of Q to itself that fixes Q pointwise.

First, we need to show that σ is a homomorphism. Let a and b be elements of E, and let f(x) be a polynomial with rational coefficients such that f(a) = 0. Then, we have:

f(σ(a)) = f(σ(a)) + f(σ(b)) - f(σ(a + b)) = f(σ(a + b)) = f(σ(a) + σ(b)) = f(σ(a)) + f(σ(b))

This shows that σ is a homomorphism.

Next, we need to show that σ is bijective. To show that σ is injective, suppose that σ(a) = σ(b) for some elements a and b of E. Then, we have:

a = σ^(-1)(σ(a)) = σ^(-1)(σ(b)) = b

This shows that σ is injective.

To show that σ is surjective, let c be an element of F. Then, we have:

c = σ(σ^(-1)(c))

This shows that σ is surjective.

Finally, we need to show that σ fixes Q pointwise. Let c be an element of Q, and let f(x) be a polynomial with rational coefficients such that f(c) = 0. Then, we have:

f(σ(c)) = f(σ(c)) + f(σ(0)) - f(σ(c + 0)) = f(σ(c + 0)) = f(σ(c) + σ(0)) = f(σ(c)) + f(σ(0))

This shows that σ fixes Q pointwise.

Conclusion

In conclusion, Q-automorphisms are bijective homomorphisms from the algebraic closure of Q to itself that fix Q pointwise. They have several important properties, including fixing Q pointwise, being bijective, and being homomorphisms. Q-automorphisms are a fundamental concept in Galois theory and have several applications in number theory and algebraic geometry.

References

Lang, S. (1993). Algebra. Addison-Wesley.
Artin, E. (1957). Galois Theory. Dover Publications.
Bourbaki, N. (1998). Algebra. Springer-Verlag.

Q: What is a Q-automorphism?

A: A Q-automorphism is a bijective homomorphism from the algebraic closure of Q to itself that fixes Q pointwise. In other words, it is an automorphism of the algebraic closure of Q that leaves every element of Q unchanged.

Q: What are some examples of Q-automorphisms?

A: Some examples of Q-automorphisms include:

Identity automorphism: The identity automorphism is a Q-automorphism that maps every element of the algebraic closure of Q to itself.
Complex conjugation: Complex conjugation is a Q-automorphism that maps every complex number to its complex conjugate.
Field automorphisms: Field automorphisms are Q-automorphisms that map every element of the algebraic closure of Q to a different element of the same field.

Q: What are the properties of Q-automorphisms?

A: Q-automorphisms have several important properties, including:

Fixing Q pointwise: Q-automorphisms fix Q pointwise, meaning that they leave every element of Q unchanged.
Bijective: Q-automorphisms are bijective, meaning that they are both injective and surjective.
Homomorphism: Q-automorphisms are homomorphisms, meaning that they preserve the algebraic structure of the algebraic closure of Q.
Automorphism: Q-automorphisms are automorphisms, meaning that they are bijective homomorphisms.

Q: How do I prove that a given map is a Q-automorphism?

A: To prove that a given map is a Q-automorphism, you need to show that it is a bijective homomorphism from the algebraic closure of Q to itself that fixes Q pointwise. This involves showing that the map is injective, surjective, and preserves the algebraic structure of the algebraic closure of Q.

Q: What are some applications of Q-automorphisms?

A: Q-automorphisms have several applications in number theory and algebraic geometry, including:

Galois theory: Q-automorphisms are used to study the Galois group of a field extension, which is a fundamental concept in Galois theory.
Algebraic number theory: Q-automorphisms are used to study the properties of algebraic numbers, which are complex numbers that are roots of non-constant polynomials with rational coefficients.
Algebraic geometry: Q-automorphisms are used to study the properties of algebraic varieties, which are geometric objects that are defined by polynomial equations.

Q: Can you provide an example of a Q-automorphism?

A: Yes, here is an example of a Q-automorphism:

Let Q be the field of rational numbers, and let F be the algebraic closure of Q. Let σ be the map from F to itself that is defined by σ(x) = x^2 for all x in F. Then, σ is a Q-automorphism because it is a bijective homomorphism from F to itself that fixes Q pointwise.

Q: How do I find the Q-automorphisms of a given field extension?

A: To find the Q-automorphisms of a given field extension, you need to find all the bijective homomorphisms from the algebraic closure of Q to itself that fix Q pointwise. This involves using the properties of Q-automorphisms and the properties of the field extension to find all the possible Q-automorphisms.

Q: Can you provide a reference for Q-automorphisms?

A: Yes, here are some references for Q-automorphisms:

Lang, S. (1993). Algebra. Addison-Wesley.
Artin, E. (1957). Galois Theory. Dover Publications.
Bourbaki, N. (1998). Algebra. Springer-Verlag.

Q: What are some further reading resources for Q-automorphisms?

A: Here are some further reading resources for Q-automorphisms:

Galois Theory by David A. Cox (Cambridge University Press, 2014)
Algebraic Number Theory by Jurgen Neukirch (Springer-Verlag, 1999)
Galois Cohomology by Jean-Pierre Serre (Springer-Verlag, 2002)