Every Finite Field is Perfect and a Galois Extension: A Detailed Proof

Understanding the properties of finite fields and their relationship with perfect fields and Galois extensions is a fundamental concept in algebra, particularly in the fields of finite and Galois theory. This article explores the proof that every finite field is a perfect field and hence a Galois extension of its prime subfield.

Theoretical Background

Definition of a Finite Field: A finite field, also known as a Galois field, is a field with a finite number of elements. It is denoted as GF(q) or Fq, where q is the order of the field.

Definition of a Prime Subfield: The prime subfield of a finite field Fq is the smallest subfield containing the multiplicative identity of Fq. In the case of finite fields, the prime subfield is always isomorphic to Fp, where p is a prime number and p is the characteristic of the field.

Key Theorems and Proofs

The proof that every finite field Fq is a perfect field and a Galois extension of its prime subfield Kq relies on several key concepts and theorems in field theory:

1. Finite Fields as Polynomial Extensions

Every finite field Fq can be viewed as a finite-dimensional vector space over its prime subfield Kq. This means that Fq is an extension of Kq of some finite degree, say d.

Theorem 1: Every finite field Fq is a finite-dimensional vector space over its prime subfield Kq, and hence is a Galois extension of Kq.

Proof: Since Fq is a vector space over Kq, every element of Fq can be written as a linear combination of a basis of Fq over Kq. The size of this basis is the dimension of Fq over Kq, which is finite. Therefore, Fq is a finite extension of Kq. By definition, this implies that Fq is a Galois extension of Kq.

2. Separability of Extensions

A field extension is separable if every element of the extension field is separable over the base field. An element is separable over a field if its minimal polynomial has no repeated roots in any extension of that field.

Theorem 2: Every finite field Fq is a separable extension of its prime subfield Kq.

Proof: Let Fq be a finite field of order q and let Kq be its prime subfield of order p, where p is a prime number. Since Kq is a field of characteristic p, it is known that Kq is a perfect field. In a perfect field, every element is separable, meaning that the minimal polynomial of every element in Fq over Kq has distinct roots in some extension of Kq. This implies that Fq is a separable extension of Kq.

3. Fundamental Theorem of Galois Theory

The Fundamental Theorem of Galois Theory states that there is a one-to-one correspondence between the subgroups of the Galois group of a finite Galois extension and the intermediate fields of that extension. This correspondence respects the lattice structure of subgroups and subfields.

Theorem 3: The Galois group of a finite Galois extension is a finite group, and the Galois group of Fq over Kq is a finite group.

Proof: By the Fundamental Theorem of Galois Theory, the Galois group of a finite Galois extension is a finite group. Since Fq is a finite extension of Kq, the Galois group of Fq over Kq is a finite group. This implies that Fq is a solvable extension of Kq, as every finite group is solvable.

4. Perfect Fields

A perfect field is a field in which every element is separable over the subfield of constants. In the context of finite fields, every finite field of characteristic p is perfect because it has the property that the Frobenius endomorphism x → xq is bijective.

Theorem 4: Every finite field Fq is a perfect field.

Proof: By definition, a finite field of characteristic p is perfect because the Frobenius endomorphism x → xq is bijective. This means that every element of Fq is separable over the prime subfield Kq, making Fq a perfect field.

Summary

Combining these theorems, we can conclude that every finite field Fq is both a perfect field and a Galois extension of its prime subfield Kq. Specifically, Fq is a separable extension of Kq and the Galois group of Fq over Kq is a finite, solvable group, making Fq a perfect field and a Galois extension.

Conclusion

The proof that every finite field is perfect and a Galois extension of its prime subfield is a fundamental result in the theory of finite fields. This result has significant implications in algebra, particularly in Galois theory and the study of solvable groups. Understanding these concepts is crucial for advanced work in algebra and related fields.