ZETA FUNCTIONS. NO.2

YOSHIFUMI TSUCHIMOTO

In this lecture we define and observe some properties of conguent zeta functions.

existence of finite fields.

Lemma 2.1. For any prime number p, ℤ/pℤ is a field. (We denote it by 𝔽_{p}.)

Funny things about this field are:

Lemma 2.2. Let p be a prime number. Let R be a commutative ring which contains 𝔽_{p} as a subring. Then we have the following facts.

(1)

\underbrace{1 + 1 + ⋯ + 1}_{p-times} = 0

holds in R.

(2) For any x, y ∈ R, we have

(x + y)^{p} = x^{p} + y^{p}

We would like to show existence of “finite fields”. A first thing to do is to know their basic properties.

Lemma 2.3. Let F be a finite field (that means, a field which has only a finite number of elements.) Then:

(1) There exists a prime number p such that p = 0 holds in F.

(2) F contains 𝔽_{p} as a subfield.

(3) q = #(F) is a power of p.

(4) For any x ∈ F, we have x^{q} − x = 0.

(5) The multiplicative group (F_{q})^{×} is a cyclic group of order q − 1.

The next task is to construct such fields. An important tool is the following lemma.

Lemma 2.4. For any field K and for any non zero polynomial f ∈ K[X], there exists a field L containing L such that f is decomposed into linear factors in L.

To prove it we use the following lemma.

YOSHIFUMI TSUCHIMOTO

Lemma 2.5. For any field K and for any irreducible polynomial f ∈ K[X] of degree d > 0, we have the following.

(1) L = K[X]/(f(X)) is a field.

(2) Let a be the class of X in L. Then a satisfies f(a) = 0.

Then we have the following lemma.

Lemma 2.6. Let p be a prime number. Let q = p^{r} be a power of p. Let L be a field extension of 𝔽_{p} such that X^{q} − X is decomposed into polynomials of degree 1 in L. Then

(1)

L_{1} = {x ∈ L; x^{q} = x}

is a subfield of L containing 𝔽_{p}.

(2) L_{1} has exactly q elements.

Finally we have the following lemma.

Lemma 2.7. Let p be a prime number. Let r be a positive integer. Let q = p^{r}. Then we have the following facts.

(1) There exists a field which has exactly q elements.

(2) There exists an irreducible polynomial f of degree r over 𝔽_{p}.

(3) X^{q} − X is divisible by the polynomial f as above.

(4) For any field K which has exactly q-elements, there exists an element a ∈ K such that f(a) = 0.

In conclusion, we obtain:

Theorem 2.8. For any power q of p, there exists a field which has exactly q elements. It is unique up to an isomorphism. (We denote it by 𝔽_{q}.)

The relation between various 𝔽_{q}’s is described in the following lemma.

Lemma 2.9. There exists a homomorphism from 𝔽_{q} to 𝔽_{q′} if and only if q′ is a power of q.

Exercise 2.1. Compute the inverse of 113 in the field 𝔽_{359}.