ABSTRACT
29.1 If something is finite, even a computer could try to write down all the elements, but it quickly shows that a systematic way of denoting the elements, keeping track of both the addition and multiplication if possible, will be necessary. This is obtained by using the primitive polynomial of a finite field (Definition 29.9), and so we are able to write down complete tables for finite fields. Of course we only do this for “small” fields like F 16 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1867.tif"/> (see Example 29.21) by hand and, in general, we can achieve this by using the computer. The fundamental observation about finite fields is that there are not that many. In fact, for a given n ∈ ℕ, there is a unique, up to isomorphism, field extension of the prime field F p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1868.tif"/> , p any prime number, of dimension n (Theorem 29.7). Note that, as prophesied in the introduction to the foregoing chapter, the cyclicity of the multiplicative group of a finite field is often used; for example, to arrive at the notion of primitive polynomial we do use that fact. The situation is very clear for a finite field: first we can always identify it as some F p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1869.tif"/> , for some n ∈ ℕ, then every subfield of it is isomorphic to F p d https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1870.tif"/> with d dividing n(Proposition 29.17), the Galois group of F p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1871.tif"/> over F p https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1872.tif"/> is generated by the Frobenius morphism (Theorem 29.17) and the Galois group of F p n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1873.tif"/> over F p ≅ F Z https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275079/07289723-0468-434b-a024-679c8189ad34/content/inequ1874.tif"/> is generated by the d-th power of the Frobenius morphism (Proposition 29.17). This theory is rounded off by providing a factorization of cyclotomic polynomials over finite fields (Proposition 29.22) and this can be done effectively by using cyclotomic cosets introduced in 29.20. A very concrete application of these techniques is found in Coding Theory.
