On a Tower of Fields related to

faculteit Wiskunde en Natuurwetenschappen

On a Tower of Fields related to ​ On ​ ^p

Master Thesis Mathematics

August 2015

Student: M. Jeeninga

First Supervisor: Prof. dr. J. Top

Second Supervisor: Dr. A.V. Kiselev

On a Tower of Fields related to On _p

Mark Jeeninga Rijksuniversiteit Groningen

mark.jeeninga@gmail.com August 27, 2015

Abstract

We study an Artin-Schreier tower of fields over Fp, for p a prime. We will show that the extensions in this tower are made by adjoining a root of the first irreducible polynomial of degree p with respect to the reverse lexicographical ordering. This generalizes a tower of fields studied by J.H.

Conway and H.W. Lenstra, Jr. in the 70s. Among other things, we give an algorithm to compute the minimal polynomials of generators in this tower and we will solve a problem posed by Lenstra in “Nim Multiplication”

(1978) for the case p = 2, and prove a generalization for p > 2.

Contents

1 Historical Introduction and Motivation 4

2 Definitions 5

3 Artin-Schreier map and traces 7

4 The tower K is a tower of fields 10

5 Orderings in the tower K 11

6 A is descending on the fields in K 12

7 The kernel of A^[n] 14

8 On_p and DiMuro’s Tower 16

9 Computing Minimal Polynomials 17

10 Equivalent Towers 18

11 The Λn sequence 20

12 Tools for Computing Sequences 22

13 The i Sequence 24

14 Discussion 30

A Appendix: Sage Codes 31

1 Historical Introduction and Motivation

This introductory text contains terminology of which a great deal will be estab- lished later on. It might be unwise to try to fully understand the details, since it is merely meant as an impression of the material we are dealing with.

The starting point of our journey is the 1976 book “On Numbers and Games”

by John H. Conway. The book gives an inductive system for constructing num- bers by the use of games. The collection of all the numbers that are constructed in this fashion is denoted by No, and it generalizes more classical notions of numbers, like the real numbers (but not the complex numbers!).

The class of ordinal numbers, which also lie inside No, is denoted by On. Like all number systems, No is totally ordered and comes with addition and multi- plication, and it even turns out to be a field.

Conway then investigates what happens if a similar inductive system is used, but where the addition of two equal elements is zero. This yields a totally ordered field of characteristic 2 he calls No2. In fact, this field contains only the ordinal numbers and is therefore also called On2. The addition and multiplication on On2differ from the regular addition and multiplication on On inside No, and are known as Nim-addition and Nim-multiplication.

The definition of the Nim-operations on On2 can be phrased in terms of the minimal excluded value, denoted by mex, of a subset S of On2. The mex of S is the first element in On2 that is not contained in S. Such an element is guaranteed to exists since the ordinal numbers are well-ordered and the ordering of On and On₂coincides. We define the Nim-addition and Nim-multiplication inductively, by respectively

α + β = mex{α⁰+ β, α + β⁰} and αβ = mex{α⁰β + αβ⁰+ α⁰β⁰},

where α⁰ runs over all ordinals less than α and β⁰ over all ordinals less than β.

This defines an algorithm that defines addition and multiplication on the or- dinals, and it does this in a remarkable sense of minimality: If the ordinals preceding α form a field k that is not algebraically closed, then α is the root of, in some sense, the first possible polynomial in k[X] that has no roots in k. If k is algebraically closed, then α is transcendental over k.

In the 1978 paper “Nim multiplication”, H.W. Lenstra, Jr. asks whether there is a more general structure, turning the ordinals into a field of arbitrary char- acteristic called On_p such that is had the same sense of minimality as On₂. This was answered positive for the addition by F. Laubie in the 1999 paper “A recursive definition of p-ary addition without carry”, which constructs On_pas an abelian group of exponent p. More recently, J. DiMuro shows in “On On_p” (2015) that for a suitable multiplication this turns On_p into a field of charac- teristic p.

If we go back to On2, it so happens that for the finite positive n the finite ordinals x_n := 2²ⁿ and q_n:=Qn−1

k=02^2k =Qn−1

k=0x_k share the relation x²_n = xn+ qn.

These relations inductively define a tower of fields in On₂that forms the closure of degree 2 extensions of F2: If we let f_n(X) = X²+X +q_n, where q_n:=Qn−1

k=0x_k and we let xn be a root of fn for all n ∈ Z≥0, then the result is an infinite field that is closed under taking extensions of degree 2. Note that this ensures that the two definitions of xn and qn coincide.

DiMuro shows in [5], Theorem 3.1(7), that a similar tower of fields exists in Onp, given by fn(X) = X^p− X − q^p−1_n , using the same construction as in the case for On2. These polynomials exhibit some sense of minimality and their roots are certain ‘nice’ ordinals in Onp. We provide a more complete description in section 8, but for the full context of this statement we refer to DiMuro’s paper.

In this paper we will show that a similar tower of fields defined by X^p−X +q_n^p−1 takes first irreducible polynomials of degree p with respect to the reverse lexi- cographical ordering to produce its extensions.

After proving this result, we will look at other properties of this tower of fields, like generating minimal polynomials of certain elements, and we solve a problem posed by Lenstra in “Nim multiplication”, which, to our knowledge, hasn’t been published before. We even go further and prove a generalization.

The Artin-Schreier map fulfils a central rˆole in this paper. We will prove many properties of this map in relation to this tower of fields. We for example show that this map is descending on the fields in the tower and we characterize the kernel of any number of iterations of the map.

2 Definitions

Throughout this paper we fix p to be a prime.

Furthermore we use the notation υ^[n]:= υ ◦ · · · ◦ υ

| {z }

n times

for some map υ and n ∈ Z≥1. For our purposes we define υ^[0]:= id.

We have chosen our indices for notions such that they make sense seen as iter- ation, e.g. qi+1= qi· xi, Tri+1= Tri◦ tri, etc.

Let L be some algebra over F^p.

Define F : L → L; z 7→ z^p to be the Frobenius endomorphism for characteristic p and A : L → L; z 7→ z^p− z to be the Artin-Schreier map for characteristic p.

Note that both maps are F^p-linear. Polynomials of the form X^p− X + b ∈ L[X]

are known as Artin-Schreier polynomials, or A-polynomials.

We define the tower K = {k₀⊆ k₁⊆ · · · } as follows:

Define k0:= F^p;

Let x_i be a root of the Artin-Schreier polynomial X^p− X + q^p−1_i where qi=

i−1

Y

j=0

xj= x0x1· · · xi−1∈ ki for all i ≥ 0, then

ki+1:= ki[xi].

Note that A(x_i) = −q_i^p−1.

It is not obvious that this definition makes sense, since taking a root of X^p− X + q_i^p−1is ambiguous for a reducible polynomial. We will see however that all these polynomials are irreducible, hence the following claim:

Claim 1 For all i ≥ 0, ki is a field isomorphic to Fp^pi.

We will prove this claim in section 4, however we will assume it throughout our further definitions for the sake of notation.

Galois groups The Galois groups Gi := Gal(ki/k0) of the tower K are iso- morphic to Z/pⁱZ and are generated by F under composition.

The relative Galois groups g_i := Gal(k_i+1/k_i) of the tower K are isomorphic to Z/pZ and are generated by F^[pi]= {z 7→ z^ppi} under composition.

Norms and traces Each Galois group induces a trace and a norm map.

G_i induces the trace Tri := X

σ∈Gi

σ = X

n∈Z/pⁱZ

F^[n] = {z 7→ X

n∈Z/pⁱZ

z^pn},

which is k₀-linear, and the norm Ni:= Y

σ∈G_i

σ = Y

n∈Z/pⁱZ

F^[n]= {z 7→ Y

n∈Z/pⁱZ

z^pn} = {z 7→ z^pp

i−1 p−1 }.

Note that Tri and Ni map elements of ki to k0. Likewise, gi induces the relative trace

tri:= X

σ∈gi

σ = X

n∈Z/pZ

F^[pn]= {z 7→ X

n∈Z/pZ

z^pnpi},

which is ki-linear, and the relative norm

ni:= Y

σ∈g_i

σ = Y

n∈Z/pZ

F^[pn]= {z 7→ Y

n∈Z/pZ

z^pnpi} = {z 7→ z

ppi+1

−1 ppi−1 }.

One observes that tr_i and n_i map elements of k_i+1 to k_i.

Here is an illustration of the (relative) trace and (relative) norm maps above:

K = { k0 ⊆ k1 N₁=n₀

cc

Tr₁=tr₀

{{ ⊆ · · · ⊆ ki

N_i

OO

Tr_i

 ⊆ ki+1

ni

cc

tr_i

{{

N_i+1

OO

Tr_i+1

 ⊆ · · · }

The triangles are in fact commutative. I.e. Tri+1= Tri◦ triand Ni+1= Ni◦ ni. It is easily checked that this holds for the norm, but it may be less trivial for the trace. We will prove this right after our definitions, in section 3.

Characteristic polynomials We look at k_i as a vector space over k₀. For α ∈ k_i, the map z 7→ αz is a k₀-linear map from k_i to k_i.

The characteristic polynomial of α Φ^α_i(X) := Y

n∈Z/pⁱZ

X − F^[n](α)
∈ F^p[X].

is the characteristic polynomial of this map. If α ∈ ki\ki−1, then Φ^α_i(X) equals the minimal polynomial of α over F^p.

The polynomial Φ^α_i has degree pⁱ. The coefficient of the X^pi−1 term of Φ^α_i is equal to −Tri(α). The constant term of Φ^α_i is equal to −Ni(α).

Likewise we can look at ki+1 as a vector space over ki. For α ∈ ki+1, the map z 7→ αz is a ki-linear map from ki+1 to ki+1. The relative characteristic polynomial

φ^α_i(X) := Y

n∈Z/pZ

X − F^[pn](α)
∈ ki[X].

is the characteristic polynomial of this map. If α ∈ k_i+1\k_i, then φ^α_i(X) equals the minimal polynomial of α over k_i.

The polynomial φ^α_i has degree p. The coefficient of the X^pi−1 term of φ^α_i is equal to −tri(α). The constant term of φ^α_i is equal to −ni(α).

E.g. φ^x_iⁱ(X) = X^p− X + q^p−1_i , which implies that tri(xi) =

(1 for p = 2 0 for p 6= 2 and ni(xi) = −q^p−1_i = A(xi).

3 Artin-Schreier map and traces

In this section we will show that the (relative) trace maps we defined above are composites of the Artin-Schreier map A.

Let L be some field extending Fp. Define R := {z 7→ az 

 a ∈ F^p} ⊆ End(L, +), where End(L, +) is the ring of F^p-linear maps from L to L. The ring R is naturally isomorphic to F^p, so R is a field. Additionally, the (commutative) polynomial rings F^p[X] and R[Z] are isomorphic.

For a field k of positive characteristic, the (non-commutative) ring k[F ] plays an important rˆole in the theory of Drinfeld modules. The map F satisfies the property that F a = a^pF for an element a ∈ k. Because F ∈ End(L, +) acts trivially on R, F commutes with R and we have R[F ] a commutative ring.

Therefore the evaluation homomorphism evF: R[Z] → End(L, +); Z 7→ F is a morphism of commutative rings, and we obtain the natural isomorphism

R[Z]/ ker(ev_F) ' im(ev_F) = R[F ].

Note that F is a map acting on L and hence our definition of R[F ] depends on L. We choose not to accentuate this by some notation, but one should keep this in mind throughout the next results.

Theorem 3.1. If L is infinite, then R[F ] ' F^p[X]. If L is finite, then R[F ] ' Fp[X]/(X^m− 1), where m = [L : Fp].

Proof. Let f (Z) ∈ ker(ev_F) be a non-zero polynomial in R[Z]. By definition f (F ) is the zero-map on L. If we look at f (F ) ∈ R[F ] as being a polynomial map on L, it implies that all elements of L are a root of f (F )(X) ∈ L[X].

Let d := deg_R[Z](f ) be the degree of f in the polynomial ring R[Z]. If we see F as a polynomial map in L[X], it has degree p. Therefore the map evF sends the polynomial f in R[Z] of the degree of d to a polynomial in L[X] of degree p^d. If L is infinite, then there cannot exist a non-zero polynomial of finite degree that has all elements in L as a root. So in this case ker(ev_F) = {0} and R[F ] ' R[Z] ' F^p[X].

If L is finite, define m := [L : F^p]. The ring R[Z] is a principal ideal do- main, so the ideal ker(ev_F) is generated by one element. Let g := Z^m− 1, then ev_F(g) = F^[m]− id and all elements in L are roots of ev_F(g)(X). Hence g ∈ ker(ev_F). Since g has degree m, the degree of ev_F(g) is p^m. The field L also has p^m elements, and so all roots of evF(g) are elements in L. This shows that no non-zero polynomials of degree less than m in R[Z] can have all elements in L as its roots, after applying evF. Therefore all elements in ker(evF) are a multiple of g. I.e. ker(evF) = (g) = (Z^m− 1), which implies that R[F ] ' R[Z]/(Z^m− 1) ' Fp[X]/(X^m− 1)

E.g. If L =S∞

n=0Fp^pn, then R[F ] ' R[Z] ' Fp[X].

E.g. If L = Fp³, then R[F ] ' R[Z]/(Z³− 1) ' Fp[X]/(X³− 1).

To make the above more explicit we define the isomorphism ϕ : F^p[X]/f (X) →

R[F ] by

ϕ(a) = {z 7→ az} for a ∈ F^p and ϕ(X) = F = {z 7→ z^p},

where f (X) = 0 if L is infinite, or f (X) = X^m−1 if L is finite with m = [L : Fp].

Its inverse ϕ⁻¹ is given by ϕ⁻¹(z 7→ az) = a for a ∈ F^p and ϕ⁻¹(F ) = X.

Theorem 3.2. Tri= A^[pi−1]and that tri = A^[pi(p−1)].

Proof. Note that A = F − id ∈ R[F ]. Therefore ϕ⁻¹(A) = ϕ⁻¹(F − id) = X − 1, which implies that ϕ⁻¹(A^[n]) = (X − 1)ⁿ. We are in characteristic p, so (X − 1)^p= X^p− 1. Applying ϕ to

(X − 1)^pi−1 =X^pi− 1 X − 1 =

pⁱ−1

X

n=0

Xⁿ

gives

A^[pi−1]=

pⁱ−1

X

n=0

ϕ(X)^[n]= {z 7→

pⁱ−1

X

n=0

z^pn} = Tri. Likewise

(X − 1)^pi(p−1)= (X^pi− 1)^p−1 =X^pi+1− 1 X^pi− 1 =

p−1

X

n=0

X^npi

implies that

A^[pi(p−1)]=

p−1

X

n=0

ϕ(X)^[npi] = {z 7→

p−1

X

n=0

z^pnpi} = tr_i.

With these identities it is easy to observe that Tri+1= Tri◦ tri since Tr_i◦ tri= A^[pi−1]◦ A^[pi(p−1)]= A^{[pi−1+pi(p−1)]}= A^[pi+1−1]= Tr_i+1. The identity tr0= Tr1 implies that tr0◦ · · · ◦ tri= Tri+1 by induction.

We can also observe that tr^[p]_i = A^{[pi(p−1)][p]} = A^{[pi+1(p−1)]} = tri+1 and the following lemma:

Lemma 3.3. A^[pi]= F^[pi]− id

Proof. Using the isomorphism ϕ we can directly verify that A^[pi] = ϕ (X − 1)^pi
= ϕ(X^pi− 1) = ϕ(X^pi) − ϕ(1) = F^[pi]− id.

For those who are familiar with linearised polynomials and symbolic multipli- cation, the correspondence of R[F ] with (a quotient of) Fp[X] is a great way to see how symbolic multiplication of linearised polynomials in R[F ] is related to the multiplication of regular polynomials in (a quotient of) Fp[X]. It also trivi- alizes the basic proofs on symbolic division and symbolic factorization. More on linearised polynomials and symbolic multiplication can be found in [6], chapter 3, section 4.

4 The tower K is a tower of fields

In this section we will prove Claim 1. To do so we will prove that the polynomial X^p− X + q^p−1_i ∈ ki[X] is irreducible over kiby induction on i. This proves that the tower K is a tower of fields, and that ki' F_ppi.

We will use the following lemma:

Lemma 4.1. An Artin-Schreier polynomial X^p− X − α ∈ ki[X] is irreducible over k_i if and only if Tr_i(α) 6= 0.

Proof. Let r ∈ ki be a root of X^p− X − α, i.e. A(r) = α. Then, Tri(α) = 0 iff 0 = A^[pi−1](α) = A^[pi](r) = r^ppi− r iff r ∈ k_i. Hence Tr_i(α) 6= 0 iff r 6∈ k_i. It therefore suffices to show that Tri(q_i^p−1) 6= 0 by induction on i.

Theorem 4.2 (Proof of Claim 1). ki ' F_ppi and Tri(q_i^p−1) = (−1)ⁱ for all i ≥ 0.

Proof by induction. The claim holds for i = 0: We have Tr0(1) = 1 6= 0, and so the polynomial X^p− X + 1 ∈ Fp[X] is irreducible by Lemma 4.1.

Now suppose that Trn(q^p−1_n ) = (−1)ⁿfor all n ≤ i. This implies that kn' Fp^pn

for all n ≤ i by Lemma 4.1. The next step in the induction is Tri+1(q_i+1^p−1) = Tri◦ tri(q_i+1^p−1) = Tri(q_i^p−1· tri(x^p−1_i )).

The last equality holds since the map tr_i is k_i-linear and q_i lies in k_i.

By dividing both sides of the equation x^p_i − xi+ q_i^p−1= 0 by xi it follows that x^p−1_i = 1 − q_i^p−1x⁻¹_i . Therefore

tri(x^p−1_i ) = tri(1 − q_i^p−1· x⁻¹_i ) = tri(1) − tri(q_i^p−1· x⁻¹_i ) = −q^p−1_i tri(x⁻¹_i ).

Note that tri(1) = 0 since tri= A^[pi(p−1)] and A(1) = 0.

By dividing both sides of the identity x^p_i − xi + q^p−1_i = 0 by q_i^p−1 · x^p_i we obtain

(q⁻¹_i )^p−1− (x⁻¹_i )^p−1(q_i⁻¹)^p−1+ (x⁻¹_i )^p= 0.

This tells us that x⁻¹_i is a root of the polynomial X^p− (q⁻¹_i )^p−1X^p−1+ (q⁻¹_i )^p−1

in k_i[X]. Since the degree of this polynomial is p, it is the minimal polynomial of x⁻¹_i over k_i. Therefore tr_i(x⁻¹_i ) = (q_i⁻¹)^p−1. This implies that

tr_i(x^p−1_i ) = −q_i^p−1· tri(x⁻¹_i ) = −q_i^p−1· (q_i⁻¹)^p−1 = −1,

and hence Tri+1(q_i+1^p−1) = −Tri(q^p−1_i ) = (−1)ⁱ⁺¹ as desired. Theorem 4.1 now implies that ki+1' F_ppi+1, which completes the induction.

Remember that A(xi) = −q_i^p−1 by the definition of xi. Therefore Theorem 4.2 implies

Corollary 4.3. Tr_i A(x_i)
= (−1)ⁱ⁺¹.

Recall that A^[pi] = F^[pi]− id by Lemma 3.3, and therefore F^[pi] = id + A^[pi]. In section 3 we saw that Tri = A^[pi−1], and hence A^[pi](xi) = Tri(A(xi)) = (−1)ⁱ⁺¹. This implies that F^[pi](xi) = xi+ (−1)ⁱ⁺¹. Hence the corollary:

Corollary 4.4. A^[mpi](xi) = m · (−1)ⁱ⁺¹and F^[mpi](xi) = xi+ m · (−1)ⁱ⁺¹ for m ≥ 0.

Now that we have established that ki is isomorphic to Fp^pi, we can formulate the ki-linearity of A^[n]in a particularly nice way:

Lemma 4.5. A^[n] is ki-linear if pⁱ| n.

Proof. We know that A^[pi]= F^[pi]− id is ki-linear, and because the composition of two ki-linear maps is again ki-linear, we find that A^[mpi] is ki-linear for all m ≥ 0.

5 Orderings in the tower K

We put an ordering < on the elements F^p in the natural way.

I.e. 0 < 1 < · · · < p − 1. The induced ordering ≤ is a well-ordering.

Definition 5.1. Let d ∈ Z≥0 and write d = Pi−1

j=0d_jp^j < pⁱ with 0 ≤ d_j < p, the p-ary expansion of d . Define λd :=Qi−1

j=0x^d_j^j ∈ ki, which are known as the monomials in k_i for d ≤ pⁱ.

E.g. xi= λ_pi, x^p−1_i = λ_pi(p−1), qi= λpi −1 p−1

and q^p−1_i = λ_pi−1.

There are exactly pⁱ monomials in ki, which form a basis of ki as an F^p-vector space. This means that for every element a ∈ ki we may write

a =

pⁱ−1

X

d=0

adλd

for certain a_d∈ Fp. Note that because all the λ_d’s are unique, this gives us p^pi elements in k_i, as expected. If a_d 6= 0 we call adλ_d and λ_d respectively a term and a monomial in (the expression of) a.

Since these sums are finite, it is well-defined to talk about the largest term and largest monomial in (the expression of) a.

Furthermore, we declare λd < λe if d < e, which induces a well-ordering on the monomials in ki. It is clear that the map d 7→ λd is bijective, and order- preserving by definition.

E.g. 1 < xi< xi+1< xi+1xi< qi+2.

We extend the ordering on the monomials in ki to an ordering on all elements in ki in the following way.

Definition 5.2. Let a = Ppⁱ−1

d=0 a_dλ_d and b = Ppⁱ−1

d=0 b_dλ_d be elements in k_i. We declare a < b if there exists n ∈ Z≥0 such that a_e= b_e for all e > n, and a_n< b_n.

In other words, we compare the coefficients of the largest monomials first. This is known as the reverse lexicographical ordering on ki.

E.g. 0 < 1 < xi< xi+1< xi+1xi in ki+2.

Now that we have established an ordering on ki, we extend it further to ki[X]

to acquire the reverse lexicographical ordering on the polynomials in ki[X]:

Definition 5.3. Consider f, g ∈ ki[X]. Write f (X) = Pdeg(f )

k=0 αkX^k and g(X) = Pdeg(g)

k=0 β_kX^k, where α_k, β_k ∈ k_i. We declare f < g if there exists an n ∈ Z≥0 such that αj= βj for all j > n, and αn< βn.

Again, we compare the coefficients in front of the largest powers of X first.

E.g. Let α, β ∈ k^∗_i such that α < β, then

0 < α < X ≤ αX < βX < X²+ αX + β < X²+ βX.

All the orderings above are well-orderings, since they are defined over discrete sets and induced by well-orderings. Hence it makes sense to talk about the first element of a set with one of the orderings above.

6 A is descending on the fields in K

Remark. Note that in characteristic p we have (−1)^p = −1, even for p = 2.

Hence we don’t have to make a distinctions between p even and p odd for such expressions.

Lemma 6.1. A^[pi] Qn

k=1ak



=Qn

k=1 A^[pi](ak) + ak
− Qⁿ_k=1ak. Proof. Using Lemma 3.3 we see that

A^[pi]Yⁿ

k=1

ak



= F^[pi]Yⁿ

k=1

ak

−

n

Y

k=1

ak =

n

Y

k=1

F^[pi](ak) −

n

Y

k=1

ak

=

n

Y

k=1

A^[pi]+ id
(ak) −

n

Y

k=1

ak

=

n

Y

k=1

A^[pi](ak) + ak
−

n

Y

k=1

ak

Corollary 6.2. Lemma 6.1 directly implies the following:

i. A^[pi](aⁿ) = A^[pi](a) + a
ⁿ

− aⁿ=Pn−1 k=0

n k



a^k· A^[pi](a)^n−k;

ii. A(aⁿ) = A(a) + a
ⁿ

− aⁿ=Pn−1 k=0

n k



a^k· A(a)^n−k;

iii. A^[pi](a · b) = A^[pi](a) · A^[pi](b) + a · A^[pi](b) + A^[pi](a) · b;

iv. A(a · b) = A(a) · A(b) + a · A(b) + A(a) · b.

Lemma 6.3. A(xⁿ_i) < xⁿ_i for 0 ≤ n < p.

Proof. First we should note that A(xi) < xi, since the term −q^p−1_i = A(xi) precedes the monomial xi. Hence the claim holds for n = 1. The claim is also true for n = 0, since A(1) = 0 < 1.

Corollary 6.2ii tell us that A(xⁿ_i) = Pn−1 k=0

n k



x^k_i · A(xi)^n−k. The ordering tells us that all elements below the monomial xⁿ_i are a sum over the monomials that lie in ki+1 = ki(xi) and that do not contain a factor of xⁿ_i. Because the monomial A(xi) lies in ki, it contains no factor of xi, and so the expression of A(xⁿ_i), which lies in ki(xi), does not contain a monomial that contains a factor of xⁿ_i. Hence A(xⁿ_i) < xⁿ_i.

Corollary 6.4. From the definition of the ordering on ki+1= ki(xi), it follows directly that α · A(xⁿ_i) < xⁿ_i for all α ∈ {a ∈ ki+1| a < xi} = ki, and 0 ≤ n < p.

If β ∈ k_i^∗ we even have that α · A(xⁿ_i) < xⁿ_i < β · xⁿ_i.

Lemma 6.5. The map A is a descending map on the monomials in k_i. I.e. A(λ_d) < λ_d for 0 ≤ d < pⁱ.

Proof by induction. The claim holds for i = 0 and i = 1, which follows from Lemma 6.3.

Suppose A(λd) < λd for 0 ≤ d < pⁱ holds for i = n > 0. We will show that the claim also holds for i = n + 1. This implies that we need to show that A(λd) < λ_d for all pⁿ≤ d < pⁿ⁺¹, or likewise λ_d∈ kn+1\kn.

If pⁿ ≤ d < pⁿ⁺¹, then we can write λ_d = λ_e· x^d_nⁿ, where e := d − d_npⁿ and d_n6= 0. Note that λ_e∈ k_n.

If λe= 1, it follows from Lemma 6.3 that A(λd) = A(x^d_nⁿ) < x^d_nⁿ= λd. If λe6= 1 then A(λe) 6= 0. Corollary 6.2iv tells us that

A(λd) = A(x^d_nⁿ· λe) = A(x^d_nⁿ) · A(λe) + x^d_nⁿ· A(λe) + A(x^d_nⁿ) · λe. Corollary 6.4 tells us that

A(x^d_nⁿ) · A(λe) < x^d_n^j < x^d_n^j · A(λe) and A(x^d_nⁿ) · λe< x^e_n^j < x^d_n^j · A(λe), since both λeand A(λe) lie in kn. Note the relationship with the previous for- mula; It implies that the largest monomial in A(λd) is the largest monomial in x^e_nⁿ· A(λe).

By the induction hypothesis we may use that A(λ_e) < λ_e, since λ_e∈ kn. Note that multiplying both sides by x^d_nⁿ doesn’t alter their relative order. Hence A(λ_e) · x^d_nⁿ < λ_e· x_n^dn= λ_d, and so A(λ_d) < λ_d.

We have shown that A(λd) < λd for 0 ≤ d < pⁿ⁺¹, which completes the induction.

Theorem 6.6. The map A is a descending map on k_i^∗. I.e. A(a) < a for all a ∈ k_i^∗.

Proof. Let a ∈ k^∗_i and write a = Ppⁱ−1

d=0 adλd for ad ∈ Fp. The map A is Fp-linear, hence

A(a) = A^p

i−1

X

d=0

a_dλ_d

=

pⁱ−1

X

d=0

a_d· A(λ_d)

Let λ_e be the largest monomial in A(a). Then, by the Fp-linearity of A, there must exist a monomial λ_din the expression of a such that λ_eis in the expression of A(λ_d); Else λ_e wouldn’t occur in A(a). From Lemma 6.5 it follows that A(λ_d) < λ_d, and so λ_e< λ_d. This implies that A(a) < a.

Note that the coefficient in front of λein the expression of A(a) doesn’t play a rˆole, by checking the definition of the ordering on ki. So, even if there would be multiple monomials in the expression of a that, after applying A, contain λe, the coefficient in front of λ_ewould be at most p − 1, while (p − 1)λ_estill precedes λ_d. Theorem 6.6 implies that A is a nilpotent in the ring of Fp-linear endomor- phisms on ki. Of course we already knew this, since ker A^[pi]= ki. This is not true for the ring of F^p-linear endomorphisms over an infinite field containing ki.

Corollary 6.7. Let a ∈ k_i^∗. The largest monomial in the expression of A(a) precedes the largest monomial in the expression of a.

Proof. Continuing the proof of theorem 6.6, let λc be the largest monomial in the expression of a, then λ_d≤ λc, and thus λ_e< λ_c.

7 The kernel of A

The next statement to prove on our list is that q^p−1_i is the first element in k_i for which the trace is non-zero. This implies that X^p− X + q_i^p−1 is the first irreducible A-polynomial in ki[X].

Theorem 7.1. ker A^[n]= {α ∈ k_i| α < λ_n} for i such that n < pⁱ.

Proof by induction. We have seen that ker A^[pi−1] = ki−1, and the ordering on ki tells us that ki−1= {α ∈ ki| α < xi−1= λ_pi−1}. Hence the claim holds for n = pⁱ⁻¹.

Suppose the claim holds for some d such that pⁱ⁻¹ ≤ d < pⁱ− 1. This means that we assume ker A^[d] = {α ∈ k_i+1 | α < λd}. Theorem 6.5 tells us that A(λd) < λ_d, and so A(λ_d) ∈ ker A^[d]. This implies that λ_d∈ ker A^[d+1]. Since A is an Fp-linear map, the kernel of A^[n] is an Fp-vector space, spanned by monomials in k_i+1.

Since ker A^[d] is a subspace of ker A^[d+1] and λ_d ∈ ker A^[d+1], we have that {α ∈ ki+1 | α < λd+1} ⊆ ker A^[d+1]. We count the number of elements in both sides of the inclusion and find that they are both p^d+1. This implies that {α ∈ ki+1 | α < λd+1} = ker A^[d+1], which proves the claim.

One could argue from the results above that the reverse lexicographical ordering defined in section 5 is in some sense natural to the Artin-Schreier map.

Lemma 7.2. X^p− X + q^p−1_i is the first irreducible A-polynomial in ki[X].

Proof. By definition, an A-polynomial in ki[X] is a polynomial of the form X^p− X + α for α ∈ ki. Lemma 4.1 implies that such a polynomial is irreducible if and only if Tri(α) 6= 0. Theorem 7.1 states ker A^[n]= {α ∈ ki| α < λn}, and since A^[pi−1]= Triand λ_pi−1 = q_i^p−1we find that ker Tri= {α ∈ ki| α < q_i^p−1}.

In other words, q_i^p−1is the first element with non-zero trace, and so X^p−X+q_i^p−1 is the first irreducible A-polynomial in ki[X].

Lemma 7.3. gcd ^ppk−1

p^pl−1, p^pm− 1
= 1 for all k ≥ l ≥ m.

Proof. Let d = gcd ^ppk−1

p^pl−1, p^pm − 1
. Then p^pm ≡ 1 (mod d) and ^ppk−1

p^pl−1 ≡ 0 (mod d). Observe that ^ppk−1

p^pl−1 =Pp^k−l−1

n=0 p^pln ≡ p^k−l (mod d), which implies that d | p^k−l.

Therefore d | gcd(p^k−l, p^pm− 1) = 1. Hence d = 1.

Now we prove an important property of the tower K that uniquely defines K.

Theorem 7.4. X^p− X + q_i^p−1 is the first irreducible polynomial of degree p in k_i[X].

The proof is inspired by Lemma 3.4 in [7].

Proof. By lemma 7.2 it suffices to show that there are no irreducible polyno- mials in k_i[X] preceding the first A-polynomial. These polynomials are of the form X^p+ mX + α, where m < −1 and α ∈ k_i.

To show that X^p+ mX + α is reducible for the given conditions, it is enough to show that the map l : z 7→ z^p+ mz is a bijection on k_i. This will then imply that there exists an element β ∈ k_i such that l(β) = −α. I.e. β is a root of X^p+ mX + α.

The map l is F^p-linear, hence a linear map on kiseen as a vector space over F^p. Let r ∈ F^p be a root of the polynomial X^p−1+ m ∈ F^p[X]. Since m 6= −1, we know that 1 6= r^p−1= −m ∈ F^p, so r 6∈ F^p and Ord(r) | (p − 1)². We will prove that r 6∈ ki:

Suppose r ∈ ki, then also Ord(r) | p^pi− 1. Therefore

Ord(r) | gcd p^pi− 1, (p − 1)²
= (p − 1) gcd ^ppi_p−1⁻¹, p − 1
.

Then by lemma 7.3, gcd(^p_p−1^pi−1, p − 1) = 1 and whence Ord(r) | p − 1. But that means that r ∈ Fp, which is a contradiction. So r 6∈ k_i, which shows that X^p−1+ m has no roots in k_i.

Thus l(X) = X^p+ mX = X(X^p−1+ m) has no roots in k_i other than zero. Let a, b ∈ ki such that l(a) = l(b), then 0 = l(a) − l(b) = l(a − b) implies that a = b since l is F^p-linear. The map l is therefore an injective map from ki to ki. It is even surjective since ki is finite, hence a bijection. This proves the claim.

The property proved in 7.4 uniquely defines K in the sense that all the extensions in K are made by adjoining a root of the first irreducible polynomial of degree p.

8 On

and DiMuro’s Tower

In [5], Theorem 3.1(7), DiMuro shows that certain ‘nice’ ordinals in On_p form a tower of fields. It satisfies some sense of minimality, just like K. I will try to explain this construction, without going into too much detail.

The field Onp is the class of all ordinals that forms a field of characteristic p. It contains the field F^p, the algebraic closure of F^p. We say that an ordinal in Onpis a field if the ordinals preceding this ordinal form a field. We will only consider the ordinals that form a field that lies inside F^p.

Let the ordinal α be a field, then the smallest ordinal that is a field in Onp

extending the field below α, i.e. the next field in Onp, is the field generated by adjoining α. This field corresponds with the ordinal β, which satisfies the equation βⁿ= f (β), where f is a polynomial over the field below β that is the first polynomial (of order less than n) such that Xⁿ− f (X) is irreducible, and n is chosen to be as small as possible. This process constructs a tower of fields that is closed under taking extensions of index 2, then 3, then 5, etc. until we have had all primes and we end up with F^p.

We put the reverse lexicographical ordering on the elements in these fields in the same way we did for K. This ordering coincides with the natural ordering of the ordinals.

We define ˜K = {˜k0 ⊆ ˜k1 ⊆ . . . } to be the tower of fields over Fp such that all extensions are made as is done in Onp with n = p, as described above. This tower is closely related to K, as we will see in a moment.

If we tensor this tower over F^p with the smallest field in Onp that is closed under taking extensions of degree less than p, we end up with the same tower DiMuro describes in [5], Theorem 3.1(7).

We claim that the fields in ˜K are constructed in the following way:

Define ˜k₀:= Fp;

Let ˜xi be a root of the Artin-Schreier polynomial X^p− X − ˜q^p−1_i where ˜q_i=

i−1

Y

j=0

˜

x_j= ˜x₀x˜₁· · · ˜x_i−1∈ ˜k_i for all i ≥ 0, then k˜_i+1:= ˜k_i[˜x_i].

This is indeed a well-defined tower of fields, which can be checked by following the same steps as in section 4. The only difference with K is the sign of the constant term in the A-polynomial. In fact, all results for K can also be applied to ˜K.

It’s not too difficult to alter the results involving K to see that the extension from ˜kito ˜ki+1in ˜K are made by adjoining a root of the polynomial X^p− f (X),

where f is the first polynomial where f (X) is the first polynomial over k_i such that X^p− f (X) is irreducible. Namely, a polynomial X^p− α is always reducible over the field of characteristic p that contains α. The next polynomials to con- sider are the polynomials of the form X^p− X − α for α ∈ ˜ki, the A-polynomials over ˜ki, and we have already seen that these can be irreducible. Just like we deduce from Theorem 7.1, we can show that q_i^p−1 is the first monomial with non-zero trace. Hence Lemma 4.1 implies X^p− X − q_i^p−1 is irreducible over ˜ki. By definition, A(˜x_i) = ˜x^p_i − ˜x_i = ˜q_i^p−1 and ˜q_i+1 = ˜x_i· ˜q_i for all i ≥ 0. We can use this to find that

˜

q^p−1_i+1 = ˜q_i^p−1x˜^p−1_i = (˜x^p_i − ˜x_i)˜x^p−1_i = ˜x^2p−1_i − ˜x^p_i.

Hence, xi+1 is root of the polynomial X^p− X − x^2p−1_i + x^p_i for all i ≥ 0.

If we define F (X, Y ) = X^p− X − Y^2p−1+ Y^p, we can inductively define ˜xito be a root of F (X, ˜xi−1) for positive i, where ˜x0is a root of X^p− X − 1. Therefore F (˜xi, ˜xi−1) = 0 for all i > 0.

We can perform the same trick in K: Since A(xi) = x^p_i − xi = −q_i^p−1 for all i, we find that

−q^p−1_i = −q^p−1_i−1x^p−1_i−1 = (x^p_i−1− xi−1)x^p−1_i−1 = x^2p−1_i−1 − x^p_i−1.

This implies that x_i can also be defined inductively as a root of F (X, x_i−1) for positive i, but now x₀ is a root of X^p− X + 1. From this we see that x₀ is a conjugate of −˜x₀.

The relationship between the towers K and ˜K is therefore clear: If p = 2 we have that x0 and ˜x0 are conjugates, and so the towers are equal (up to conjugation of the generators xi and ˜xi).

If p is odd, then xi = −˜xi and F (xi+1, xi) = 0 implies that F (−xi+1, −xi) = F (−xi+1, ˜xi) = 0, since f is an odd polynomial for odd p, and so −˜xi+1 is a conjugate of xi+1. This shows that every property in K also holds for ˜K if we substitute −˜xi for xi.

9 Computing Minimal Polynomials

In this section we will show that the polynomial F (X, Y ) = X^p−X −Y^2p−1+Y^p over F^p allows an algorithm to compute minimal polynomials over ki of xi+n

for non-negative i and n. We have already seen that F (X, xi) is the minimal polynomial of xi+1 over ki+1. We define F1(X, Y ) := F (X, Y ) and the algo- rithm will compute the sequence of polynomials Fn(X, Y ) ∈ F^p[X, Y ] such that F_n(X, x_i) is the minimum polynomial of x_i+n over k_i+1. The following algo- rithm computes F_n+1given F_n as input:

Input: Fn

Output: Fn+1, Gn+1

define Gn+1(X, Y^p− Y ) :=Q

m∈FpFn(X, Y + m) define Fn+1(X, Y ) := Gn+1(X, Y^2p−1− Y^p) return Fn+1(X, Y ), Gn+1(X, Y )

Additionally, the algorithm generates polynomials G_n ∈ Fp[X, Y ] for n > 1, which we extend by defining G₁(X, Y ) := X^p− X − Y .

The proof why this algorithm works goes by induction on n:

Suppose that F_n(X, x_i) is the minimal polynomial of x_i+n over k_i+1 for i > 0.

This means that Fn(X, xi) has degree pⁿ, since [ki+n(xi+n) : ki+1] = pⁿ. The polynomial Q

m∈FpFn(X, Y + m) is invariant under the transformation Y 7→ Y + 1, by construction. Therefore it is a polynomial in the variable U := Y^p− Y , and so we write Q

m∈FpFn(X, Y + m) = Gn+1(X, U ) for some Gn+1∈ Fp[X, U ].

By the induction hypothesis, the minimal polynomial of xi+n over ki+1 is Fn(X, xi), and so Fn(xi+n, xi) = 0 implies that Gn+1(xi+n, x^p_i − xi) = 0. Note that this is true for all i ≥ 0, but because we chose i > 0, we also have that F (xi, xi−1) = 0, and so x^p_i − xi = x^2p−1_i−1 − x^p_i−1. This implies that also Gn+1(xi+n, x^2p−1_i−1 − x^p_i−1) = 0. Hence Fn+1(xi+n, xi−1) = 0. Now since the de- gree of Fn(X, xi) is pⁿ, we have that Fn+1(X, xi−1) has degree pⁿ⁺¹. This degree corresponds with the index of the field extension [k_i+n(x_i+n) : k_i] = pⁿ⁺¹, and so F_n+1(X, x_i−1) is the minimal polynomial of x_i+nover k_i.

So far we can compute all minimum polynomials of x_i+n over k_i for i > 0 and n ≥ 0, but not yet over k₀. To do this we remember that x^p₀− x₀ = −1, and since we had Gn+1(xi+n, x^p_i − xi) = 0 for all i ≥ 0, it implies that for i = 0 we have Gn+1(xn, −1) = 0. Because Gn+1(X, −1) ∈ F^p[X] has degree pⁿ⁺¹, we find that Gn+1(X, −1) is the minimal polynomial of xn over k0.

E.g. For p = 3 we have

G₂(X, Y ) = X⁹+ X⁶Y + X⁴Y + X³(Y²− 1) + X²Y − XY²− Y⁵ and

F₂(X, Y ) = (X⁹− X³) + X⁶(Y³− Y )Y²+ X⁴(Y³− Y )Y²

+ (X³− X)(Y³− Y )²Y⁴+ X²(Y³− Y )Y²− (Y³− Y )⁵Y¹⁰. A Sage implementation of the algorithm is included in the Appendix.

10 Equivalent Towers

If we let h ∈ F^p[X] and f ∈ F^p[X, Y ], we define the (h, f )p-tower to be the tower of fields

{Fp⊆ Fp(ξ0) ⊆ F^p(ξ0, ξ1) ⊆ · · · ⊆ F^p(ξ0, ξ1, . . . , ξn) ⊆ · · · }