Zp , Qp , AND THE RING OF WITT VECTORSYOSHIFUMI TSUCHIMOTO

Playing with “digits in base n”
You should know that every positive integer may be written in decimal notation:(531)10 = 5 × 102 + 3 × 101 + 1 × 100 .
Similarly, given any integer (“base”) b ≥ 2, we may write a number asa string of digits in base n. For example, we have(531)10 = 1 × 73 + 3 × 72 + 5 × 7 + 6 × 1 = (1356)7 .
Similarly, we have(531)10 = (1356)7 = (1023)8 = 10000100112 = (213)16 .
You may also probably know (repeating) decimal expresions of positive rational numbers.
(531.79)10 = 5 × 102 + 3 × 101 + 1 × 100 + 7 × 10−1 + 9 × 10−2 .
(531.79)10 = (1356.5̇346̇)7 = (1023.624̇3656050753412172702̇)8
Now let us reverse the order of digits. Namely, we employ a notation
like this1:[97.135]10 = (531.79)10[0.135]10 = (531)10[123.456]10 = (654.321)10...
Let us do some calculation with the above notation:[0.1]10 + [0.9]10 = [0.01]10[0.1]10 × [0.9]10 = [0.9]10[0.01]10 × [0.09]10 = [0.009]10You may recognize curious rules of computations. This curious notation
will lead you to a new world called “the world of addic numbers”.
Exercise 0.1. Compute[0.12345]8 + [0.75432]8with our curious notation. Then do the same computation in the usual
digital notation in base 10.
Lemma 0.1. For any prime number p, Z/pZ is a field. (We denote
it by Fp .)Lemma 0.2. Let p be a prime number. Let R be a commutative ringwhich contains Fp as a subring. Then we have the following facts.
1This is our private notation.

YOSHIFUMI TSUCHIMOTO(1)
1| + 1 +{z· · · + 1} = 0p-times

holds in R.
(2) For any x, y ∈ R, we have(x + y)p = xp + y p
0.1. Finite fields. In this subsection we study some basic properties
on finite fields. A good account can be found in [2]. Also, there is abrief explanation in [1] available on the net.
Lemma 0.3. 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 Fp as a subfield.
(3) q = #(F ) is a power of p.
(4) For any x ∈ F , we have xq − x = 0.
(5) The multiplicative group (Fq )× is a cyclic group of order q − 1.
The next task is to construct such fields. An important tool is thefollowing lemma.
Lemma 0.4. 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.
Lemma 0.5. 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 0.6. Let p be a prime number. Let q = pr be a power of p.
Let L be a field extension of Fp such that X q − X is decomposed intopolynomials of degree 1 in L. Then(1)
L1 = {x ∈ L; xq = x}
is a subfield of L containing Fp .
(2) L1 has exactly q elements.
Finally we have the following lemma.
Lemma 0.7. Let p be a prime number. Let r be a positive integer.
Let q = pr . 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 Fp .
(3) X q − X is divisible by the polynomial f as above.
(4) For any field K which has exactly q-elements, there exists anelement a ∈ K such that f (a) = 0.
In conclusion, we obtain:
Theorem 0.8. For any power q of p, there exists a field which has
exactly q elements. It is unique up to an isomorphism. (We denote itby Fq .)Zp , Qp , AND THE RING OF WITT VECTORSThe relation between various Fq ’s is described in the following lemma.
Lemma 0.9. There exists a homomorphism from Fq to Fq′ if andonly if q ′ is a power of q.
References[1] James S. Milne, Fields and galois theory (v4.61), 2020, Available atwww.jmilne.org/math/, p. 138.
[2] J. P. Serre, Cours d’arithmétique, Presses Universitaires de France, 1970.