Witt#5a: Polynomials that can be written as big wn

Witt vectors. Part 1 Michiel Hazewinkel

Sidenotes by Darij Grinberg

Witt#5a: Polynomials that can be written as big w_n [completed, not proofread]

The point of this note is to generalize the property of p-adic Witt polynomials that appeared as Theorem 1 in [2] to big Witt polynomials.

First, let us introduce the notation that we are going to use.

Definition 1. Let P denote the set of all primes. (A prime means an integer n > 1 such that the only divisors of n are n and 1. The word

”divisor” means ”positive divisor”.)

Definition 2. We denote the set {0,1,2, ...} by N, and we denote the set {1,2,3, ...} by N+. (Note that our notations conflict with the notations used by Hazewinkel in [1]; in fact, Hazewinkel uses the letter Nfor the set {1,2,3, ...}, which we denote by N+.)

Definition 3. Let Ξ be a family of symbols. We consider the polynomial ring Q[Ξ] (this is the polynomial ring over Q in the indeterminates Ξ; in other words, we use the symbols from Ξ as variables for the polynomials) and its subring Z[Ξ] (this is the polynomial ring over Z in the indetermi- nates Ξ). ¹. For any n∈N, let Ξⁿ mean the family of the n-th powers of all elements of our family Ξ (considered as elements of Z[Ξ]) ². (There- fore, whenever P ∈ Q[Ξ] is a polynomial, then P (Ξⁿ) is the polynomial obtained from P after replacing every indeterminate by its n-th power.³) Note that if Ξ is the empty family, then Q[Ξ] simply is the ring Q, and Z[Ξ] simply is the ringZ.

Definition 4. For any integerm, the set{n ∈N⁺ |(n|m)}will be denoted byN^|m. This set N^|m is the set of all divisors of m.

Definition 5. If N is a set, we shall denote by X_N the family (X_n)_n∈N of distinct symbols. Hence, Z[X_N] is the ring Z

(X_n)_n∈N

(this is the polynomial ring overZin|N|indeterminates, where the indeterminates are labelledX_n, wherenruns through the elements of the setN). For instance, Z

X_N+

is the polynomial ring Z[X₁, X₂, X₃, ...] (since N⁺ ={1,2,3, ...}), and Z

X{1,2,3,5,6,10}

is the polynomial ring Z[X₁, X₂, X₃, X₅, X₆, X₁₀].

If A is a commutative ring with unity, if N is a set, if (x_d)_d∈N ∈ A^N is a family of elements ofA indexed by elements of N, and ifP ∈Z[X_N], then

1For instance, Ξ can be (X0, X1, X2, ...), in which case Z[Ξ] means Z[X0, X1, X2, ...].

Or, Ξ can be (X0, X1, X2, ...;Y0, Y1, Y2, ...;Z0, Z1, Z2, ...), in which case Z[Ξ] means Z[X0, X1, X2, ...;Y0, Y1, Y2, ...;Z0, Z1, Z2, ...].

2In other words, if Ξ = (ξi)_i∈I, then we define Ξⁿ as (ξ_iⁿ)_i∈I. For instance, if Ξ = (X0, X1, X2, ...), then Ξⁿ = (X₀ⁿ, X₁ⁿ, X₂ⁿ, ...). If Ξ = (X₀, X₁, X₂, ...;Y₀, Y₁, Y₂, ...;Z₀, Z₁, Z₂, ...), then Ξⁿ = (X₀ⁿ, X₁ⁿ, X₂ⁿ, ...;Y₀ⁿ, Y₁ⁿ, Y₂ⁿ, ...;Z₀ⁿ, Z₁ⁿ, Z₂ⁿ, ...).

3For instance, if Ξ = (X0, X1, X2, ...) and P(Ξ) = (X0+X1)² −2X3 + 1, then P(Ξⁿ) = (X₀ⁿ+X₁ⁿ)²−2X₃ⁿ+ 1.

we denote byP (x_d)_d∈N

the element ofAthat we obtain if we substitutex_d forXdfor everyd∈N into the polynomialP. (For instance, ifN ={1,2,5}

and P = X₁² +X₂X₅ −X₅, and if x₁ = 13, x₂ = 37 and x₅ = 666, then P (x_d)_d∈N

= 13²+ 37·666−666.)

Definition 6. For anyn ∈N+, we define a polynomial w_n ∈Z h

X_N|ni by w_n=X

d|n

dX_d^nd.

Hence, for every commutative ringAwith unity, and for any family (x_k)_k∈

N|n ∈ A^N|n of elements of A, we have

w_n (x_k)_k∈

N|n

=X

d|n

dxⁿ_d^d.

The polynomials w₁, w₂, w₃, ... are called the big Witt polynomials or, simply, the Witt polynomials.

Caution: These polynomialsw₁, w₂, w₃, ...are referred to asw₁, w₂, w₃, ...

most of the time in [1] (beginning with Section 9). However, in Sections 5-8 of [1], Hazewinkel uses the notationsw₁, w₂, w₃, ...for some different poly- nomials (the so-calledp-adic Witt polynomials, defined by formula (5.1) in [1]), which arenot the same as our polynomials w₁, w₂, w₃, ...(though they are related to them: namely, the polynomial denoted byw_k in Sections 5-8 of [1] is the polynomial that we are denoting by w_p^k here after a renaming of variables; on the other hand, the polynomial that we call w_k here is something completely different).

Definition 7. Letn∈Z\ {0}. Let p∈P. We denote byvp(n) the largest nonnegative integer m satisfying p^m |n. Clearly, p^vp(n) |n and v_p(n)≥0.

Besides, v_p(n) = 0 if and only if p-n.

We also set vp(0) = ∞; this way, our definition of vp(n) extends to all n∈Z (and not only to n ∈Z\ {0}).

Definition 8. Letn∈N+. We denote by PFn the set of all prime divisors ofn. By the unique factorization theorem, the set PFnis finite and satisfies n= Q

p∈PFn

p^vp(n).

Let us now formulate our main result:

Theorem 1. Let Ξ be a family of symbols. Let τ ∈Z[Ξ] be a polynomial.

Letm ∈N. Then, the following two assertions A and B are equivalent:

Assertion A: There exists a family (τ_d)_d∈

N|m ∈ (Z[Ξ])^N|m such that τ = w_m

(τ_d)_d∈

N|m

.

Remarks: 1) Here, ∂

∂ξτ means the derivative of the polynomial τ ∈ Z[Ξ]

with respect to the variable ξ.

2) Theorem 1 makes sense even in the case when Ξ is the empty family (in this case, the AssertionB is vacuously true (since noξ ∈Ξ exists), and therefore Theorem 1 claims that in this case Assertion A is true as well;

see Corollary 3 for details).

Before we come to proving this theorem, let us remark why exactly this Theorem 1 generalizes the Theorem 1 of [2]. In fact, if p is a prime and n∈N, then the big Witt polynomial w_pⁿ (the one that we have defined above, not the one called w_pⁿ in [2]) is

w_pⁿ =X

d|pⁿ

dX_d^pnd = X

d∈N|pn

dX_d^pnd

=

n

X

k=0

p^kX_p^pk^npk since N|pⁿ =

p⁰, p¹, ..., pⁿ (becausep is a prime)

=

n

X

k=0

p^kX_p^pk^n−k since pⁿp^k =p^n−k ,

and therefore this polynomial w_pⁿ is equal to the polynomial denoted by w_n in [2]⁴, up to a renaming of variables (in fact, if we rename the variable X_p^k as X_k for every k ∈ N, then w_pⁿ =

n

P

k=0

p^kX_p^pk^n−k becomes w_pⁿ =

n

P

k=0

p^kX_k^pn−k, which is exactly the formula defining wn in [2]). Hence, in the case when m = pⁿ for a prime p and an integer n∈N, and when Ξ = (X₀, X₁, X₂, ...), the AssertionsA and Bof our Theorem 1 are identical with the Assertions A andB of the Theorem 1 in [2], and therefore our Theorem 1 yields the Theorem 1 in [2].

Before we come to the proof of Theorem 1, let us state a simple fact: If Ξ is a family of symbols, then

∂

∂ξP^g =gP^g−1· ∂

∂ξg

(1) for every ξ ∈ Ξ, every P ∈ Z[Ξ] and every positive integer g. (This can be proven either using the chain rule for differentiation, or by induction on g using the Leibniz rule.)

Proof of Theorem 1. Proof of the implication A =⇒ B: Assume that the As- sertion A holds. Then, there exists a family (τ_d)_d∈

N|m ∈ (Z[Ξ])^N|m such that τ = w_m

(τ_d)_d∈

N|m

. Hence,

τ =w_m (τ_d)_d∈

N|m

=X

d|m

dτ_d^md,

4Let us remind ourselves once again that this isnot the polynomial that we callwnin this present note.

and thus every ξ ∈Ξ satisfies

∂

∂ξτ = ∂

∂ξ X

d|m

dτ_d^md =X

d|m

d ∂

∂ξτ_d^md

| {z }

=(md)τ_d^md−1·





∂

∂ξ^τd



 (by (1), applied toP=τdandg=md)

=X

d|m

d(md)

| {z }

=m

τ_d^md−1· ∂

∂ξτ_d

=mX

d|m

τ_d^md−1· ∂

∂ξτ_d

| {z }

∈Z[Ξ]

∈mZ[Ξ],

so that Assertion B holds. Thus, we have shown that whenever Assertion A holds, Assertion B must hold as well. This proves the implication A=⇒ B.

Proof of the implication B =⇒ A: Let us assume that Assertion B holds. Thus, we have ∂

∂ξτ ∈mZ[Ξ] for every ξ∈Ξ. If we rename ξ as η here, this rewrites as follows:

We have ∂

∂ητ ∈mZ[Ξ] for everyη∈Ξ.

Let us introduce some notation:

For every family j ∈ N^Ξ and every ξ ∈ Ξ, let us denote by j_ξ the ξ-th member of the family j. Then, every family j ∈N^Ξ satisfiesj = (j_ξ)_ξ∈Ξ.

Let N^Ξfin denote the set

j ∈N^Ξ | only finitely many ξ ∈Ξ satisfyj_ξ 6= 0 . For every j ∈N^Ξfin, let Ξ^j denote the monomial Q

ξ∈Ξ

ξ^jξ. For every polynomialP ∈Z[Ξ], let coeffjP denote the coefficient of P before this monomial Ξ^j. Then, every polynomial P ∈Z[Ξ] satisfies

P = X

j∈N^Ξ_fin

coeff_jP ·Ξ^j. (2)

(This sum P

j∈N^Ξ_fin

coeff_jP ·Ξ^j has only finitely many nonzero summands, since every polynomial has only finitely many nonzero coefficients.)

For every n ∈Nand every j ∈N^Ξfin, let us denote by nj ∈N^Ξfin the family (nj_ξ)_ξ∈Ξ. Clearly, 1j = (1j_ξ)_ξ∈Ξ = (j_ξ)_ξ∈Ξ =j.

If k ∈ N^Ξfin and n ∈ N, then we write n | k if and only if (n|k_ξ for every ξ ∈Ξ).

If k ∈ N^Ξfin and n ∈ N are such that n | k, then we can define a family kn ∈ N^Ξfin

by kn = k_ξ

n

ξ∈Ξ

(indeed, k_ξ

n ∈ N for every ξ ∈ Ξ, since n | k yields n | k_ξ). This family kn clearly satisfies n(kn) =

nk_ξ

n

ξ∈Ξ

= (kξ)_ξ∈Ξ = k. Also, it is obvious that k1 =

k_ξ 1

ξ∈Ξ

= (k_ξ)_ξ∈Ξ =k.

Now, according to (2), our polynomial τ satisfies τ = P

j∈N^Ξfin

coeff_jτ ·Ξ^j. Thus, for

every η ∈Ξ, we have

∂

∂ητ = ∂

∂η X

j∈N^Ξfin

coeffjτ ·Ξ^j = X

j∈N^Ξfin

coeffjτ · ∂

∂ηΞ^j = X

j∈N^Ξfin

coeffjτ · ∂

∂η



η^jη Y

ξ∈Ξ\{η}

ξ^jξ







since Ξ^j =Y

ξ∈Ξ

ξ^jξ =η^jη Y

ξ∈Ξ\{η}

ξ^jξ





= X

j∈N^Ξ_fin

coeff_jτ ·

∂

∂ηη^jη

| {z }

=







j_ηη^jη−1, if j_η >0;

0, ifj_η = 0

Y

ξ∈Ξ\{η}

ξ^jξ = X

j∈N^Ξ_fin

coeff_jτ·

j_ηη^jη−1, if j_η >0;

0, if j_η = 0

Y

ξ∈Ξ\{η}

ξ^jξ

= X

j∈N^Ξfin; jη>0

coeff_jτ·

j_ηη^jη−1, if j_η >0;

0, if j_η = 0

| {z }

=jηη^jη−1,since jη>0

Y

ξ∈Ξ\{η}

ξ^jξ + X

j∈N^Ξfin; jη=0

coeff_jτ ·

j_ηη^jη−1, if j_η >0;

0, ifj_η = 0

| {z }

=0,sincejη=0

Y

ξ∈Ξ\{η}

ξ^jξ

= X

j∈N^Ξ_fin; jη>0

coeff_jτ·j_ηη^jη−1 Y

ξ∈Ξ\{η}

ξ^jξ + X

j∈N^Ξ_fin; jη=0

coeff_jτ ·0 Y

ξ∈Ξ\{η}

ξ^jξ

| {z }

=0

= X

j∈N^Ξ_fin; jη>0

coeff_jτ·j_ηη^jη−1 Y

ξ∈Ξ\{η}

ξ^jξ.

(3) Now, define a map

F :

j ∈N^Ξfin | j_η >0 →N^Ξfin defined by F(j) =

j_ξ, if ξ 6=η;

j_η −1, if ξ =η

ξ∈Ξ

for every j ∈N^Ξfin satisfying j_η >0.

This map F is a bijection (in fact, this map leaves all members of the family j fixed, except of the η-th member, which is reduced by 1). By the definition of F, every j ∈N^Ξfin satisfying j_η >0 is mapped to F (j) =

j_ξ, if ξ6=η;

jη −1, if ξ =η

ξ∈Ξ

. Hence, for every ξ ∈ Ξ, we have (F(j))_ξ =

j_ξ, if ξ 6=η;

jη −1, if ξ =η . In other words, (F (j))_ξ = j_ξ if

ξ6=η, and (F (j))_η =j_η −1 (sinceη=η). Using these two equations, (3) becomes

∂

∂ητ = X

j∈N^Ξ_fin; jη>0

coeff_jτ

| {z }

=coeff_F−1(F(j))τ

· j_η

|{z}

=(jη−1)+1

=(F(j))_η+1

η^jη−1

| {z }

=η^(F(j))η (since (F(j))_η=jη−1)

Y

ξ∈Ξ\{η}

ξ^jξ

|{z}

=ξ^(F(j))ξ (sinceξ∈Ξ\{η}

yieldsξ6=ηand thus (F(j))_ξ=jξ)

= X

j∈N^Ξ_fin; jη>0

coeff_F⁻¹_(F(j))τ ·

(F (j))_η+ 1

η^(F(j))η Y

ξ∈Ξ\{η}

ξ^(F(j))ξ

= X

j∈N^Ξ_fin

coeff_F⁻¹_(j)τ·(j_η+ 1)η^jη Y

ξ∈Ξ\{η}

ξ^jξ

| {z }

=Q

ξ∈Ξ

ξ^jξ=ξ^j

here we substituted F(j) forj in the sum, since the map F is a bijection

= X

j∈N^Ξ_fin

coeff_F⁻¹_(j)τ·(j_η+ 1)ξ^j.

Hence, for everyj ∈N^Ξfin, we have coeff_j ∂

∂ητ

= coeff_F⁻¹_(j)τ·(j_η+ 1). But we must have coeff_j

∂

∂ητ

∈mZ (since ∂

∂ητ ∈mZ[Ξ]). Thus,

coeff_F⁻¹_(j)τ ·(j_η+ 1) ∈mZ for every j ∈N^Ξfin. (4) Thus, every j ∈N^Ξfin and everyη∈Ξ satisfy

coeff_jτ ·j_η ∈mZ (5)

(since (4), applied to F (j) instead of j, yields coeff_F⁻¹_(F(j))τ ·

(F (j))_η+ 1

∈ mZ, which simplifies to coeff_jτ ·j_η ∈ mZ because F⁻¹(F (j)) and because (F (j))_η

| {z }

=jη−1

+1 = (j_η −1) + 1 =j_η).

Now we recall the following result from [4]:

Theorem 2. Let Ξ be a family of symbols. Let N be a nest⁵, and let (b_n)_n∈N ∈ (Z[Ξ])^N be a family of polynomials in the indeterminates Ξ.

Then, the two following assertionsC_Ξ and D_Ξ are equivalent:

Assertion C_Ξ: Every n∈N and everyp∈PFn satisfies b_np(Ξ^p)≡b_nmodp^vp(n)Z[Ξ].

Assertion DΞ: There exists a family (xn)_n∈N ∈(Z[Ξ])^N of elements ofZ[Ξ]

such that

b_n =w_n (x_k)_k∈N

for every n∈N .

6

This Theorem 2 is part of Theorem 13 in [4] (which claims that the assertions C_Ξ, DΞ,D_Ξ⁰ ,EΞ,E_Ξ⁰,FΞ, GΞ and HΞ are equivalent, whereCΞ and DΞ are our assertionsCΞ

and D_Ξ, while D⁰_Ξ, E_Ξ, E_Ξ⁰, F_Ξ, G_Ξ and H_Ξ are some other assertions). Hence, for the proof of Theorem 2, we refer the reader to [4].

Now, let us continue with the proof of Theorem 1:

Let N =N|m. Then, every element n of N is a divisor of m, and hence mn ∈N for every n ∈N.

We are going to apply Theorem 2 to the family (b_n)_n∈N ∈(Z[Ξ])^N defined by b_n= X

j∈N^Ξfin; (mn)|j

coeff_jτ·Ξ^j(mn) for every n ∈N.

Let n ∈ N and every p ∈ PFn. The polynomial bnp(Ξ^p) is the polynomial obtained from b_np after replacing every indeterminate by its n-th power. Since

b_np = X

j∈N^Ξfin; (m(np))|j

coeff_jτ · Ξ^j(m(np))

| {z }

=Q

ξ∈Ξ

ξ^(j(m(np)))ξ

= X

j∈N^Ξfin; (m(np))|j

coeff_jτ·Y

ξ∈Ξ

ξ^(j(m(np)))ξ,

it must therefore be b_np(Ξ^p) = X

j∈N^Ξ_fin; (m(np))|j

coeff_jτ·Y

ξ∈Ξ

(ξ^p)^(j(m(np)))ξ = X

j∈N^Ξ_fin; (m(np))|j

coeff_jτ ·Y

ξ∈Ξ

(ξ^p)^jξn(mp)

| {z }

=ξ^p·jξn(mp)=ξ^jξnm

since (j(m(np)))_ξ = j_ξ

(mn)p =j_ξn(mp)

= X

j∈N^Ξfin; (m(np))|j

coeff_jτ·Y

ξ∈Ξ

ξ^jξnm = X

j∈N^Ξfin; (pmn)|j

coeff_jτ ·Y

ξ∈Ξ

ξ^jξnm (6)

(since m(np) = pmn). Now, let us prove that

every j ∈N^Ξfin which satisfies (mn)|j and (pmn)-j must satisfy coeff_jτ ≡0 modp^vp(n)Z[Ξ]. (7)

In fact, let j ∈ N^Ξfin be such that (mn) | j and (pmn) - j. We have to prove that coeff_jτ ≡ 0 modp^vp(n)Z[Ξ]. Assume, for the sake of contradiction, that the op- posite holds, i. e. that coeffjτ 6≡ 0 modp^vp(n)Z[Ξ]. Then, p^vp(n) - coeffjτ, so that v_p(coeff_jτ)< v_p(n). Hence,v_p(coeff_jτ)≤v_p(n)−1 (since v_p(coeff_jτ) and v_p(n) are integers). But for every η∈Ξ, the relation (5) yields m|coeff_jτ·j_η and thus

vp(m)≤vp(coeffjτ ·jη) =vp(coeffjτ)

| {z }

≤vp(n)−1

+vp(jη)≤(vp(n)−1) +vp(jη),

6Here, wn (xk)_k∈N

meanswn

(xk)_k∈

N|n

(becauseN|n is a subset ofN, since n∈N and since nis a nest).

so that

vp(jη)≥ vp(m)

| {z }

=vp((mn)·n)

=vp(mn)+vp(n)

−(vp(n)−1) =vp(mn) + 1,

and thus p^vp(mn)+1 | j_η. On the other hand, mn | j_η (since mn | j). Thus, lcm p^vp(mn)+1, mn

|j_η. But lcm p^vp(mn)+1, mn

=pmn(in fact, gcd p^vp(mn)+1, mn

= p^{vp(mn) 7}, and thus the formula lcm (a, b) = ab

gcd (a, b) (which holds for any two posi- tive integers a and b) yields lcm p^vp(mn)+1, mn

= p^vp(mn)+1·mn

p^vp(mn) = pmn).

Hence, (pmn) | j_η. Since this holds for any η ∈ Ξ, we have thus shown that (pmn)|j, contradicting our assumption that (pmn)-j. This contradiction shows that our assumption that coeff_jτ 6≡0 modp^vp(n)Z[Ξ] was wrong. Thus, (7) is proven.

Now, every n∈N and everyp∈PFn satisfy bn = X

j∈N^Ξfin; (mn)|j

coeffjτ ·Ξ^j(mn) = X

j∈N^Ξfin; (mn)|j;

(pmn)|j

coeffjτ ·Ξ^j(mn)+ X

j∈N^Ξfin; (mn)|j;

(pmn)-j

coeffjτ

| {z }

≡0 modp^vp(n)Z[Ξ]

(by (7))

·Ξ^j(mn)

≡ X

j∈N^Ξfin; (mn)|j;

(pmn)|j

coeff_jτ ·Ξ^j(mn)+ X

j∈N^Ξfin; (mn)|j;

(pmn)-j

0·Ξ^j(mn)

| {z }

=0

= X

j∈N^Ξfin; (mn)|j;

(pmn)|j

coeff_jτ ·Ξ^j(mn)

= X

j∈N^Ξfin; (pmn)|j

coeff_jτ· Ξ^j(mn)

| {z }

=Q

ξ∈Ξ

ξ^(j(mn))ξ

since for every j ∈N^Ξfin, the conditions ((mn)|j and (pmn)|j) are equivalent, because if (pmn)|j, then (mn)|j

= X

j∈N^Ξfin; (pmn)|j

coeff_jτ·Y

ξ∈Ξ

ξ^(j(mn))ξ

= X

j∈N^Ξfin; (pmn)|j

coeffjτ·Y

ξ∈Ξ

ξ^jξnm

since (j(mn))_ξ = j_ξ

mn =jξnm

=bnp(Ξ^p) modp^vp(n)Z[Ξ] (by (6)).

Hence, we have shown that every n ∈ N and every p ∈ PFn satisfies b_np(Ξ^p) ≡ b_nmodp^vp(n)Z[Ξ]. Thus, Assertion C_Ξ of Theorem 2 holds for our family (b_n)_n∈N ∈ (Z[Ξ])^N. Consequently, Assertion D_Ξ of Theorem 2 also holds for this family (since

7In fact, the number gcd p^vp(mn)+1, mn

must be a power of p (since it is a divisor of p^vp(mn)+1, andpis a prime) and a divisor ofmn, so it must be a power ofpwhich dividesmn, and thus it must bep^κ for some integerκsatisfying 0≤κ≤vp(mn). Thus, gcd p^vp(mn)+1, mn

= p^κ | p^vp(mn) (since κ ≤vp(mn)). On the other hand, p^vp(mn) | gcd p^vp(mn)+1, mn

(since

Theorem 2 states that assertions C_Ξ and D_Ξ are equivalent). In other words, there exists a family (xn)_n∈N ∈(Z[Ξ])^N of elements of Z[Ξ] such that

b_n =w_n (x_k)_k∈N

for every n∈N . Applying this to n =m, we obtain b_m = w_m (x_k)_k∈N

= w_m (x_k)_k∈

N|m

. Renaming the family (x_k)_k∈

N|m as (τ_d)_d∈

N|m, we can rewrite this as b_m =w_m (τ_d)_d∈

N|m

. Since b_m = X

j∈N^Ξ_fin; (mm)|j

coeff_jτ·Ξ^j(mm)

| {z }

=Ξ^j1=Ξ^j

= X

j∈N^Ξ_fin; (mm)|j

coeff_jτ·Ξ^j = X

j∈N^Ξ_fin

coeff_jτ ·Ξ^j

since every j ∈N^Ξfin satisfies (mm)|j, because mm= 1

=τ (by (2)), this rewrites asτ =w_m

(τ_d)_d∈

N|m

. Thus, Assertion A holds. Hence, we have derived Assertion A from Assertion B. This proves the implication B =⇒ A.

Altogether we have now proven the implications A =⇒ B and B =⇒ A. We can thus conclude that the assertionsA and B are equivalent. This proves Theorem 1.

We notice a trivial corollary from Theorem 1:

Corollary 3. Let τ ∈ Z be an integer. Let m ∈ N. Then, there exists a family (τ_d)_d∈

N|m ∈Z^N|m of integers such thatτ =w_m (τ_d)_d∈

N|m

.

Proof of Corollary 3. Let Ξ be the empty family. Then,Z[Ξ] =Z (in fact, Z[Ξ] is the ring of all polynomials in the indeterminates Ξ over Z, but Ξ is the empty family, and polynomials in an empty family of indeterminates overZare the same as integers).

Clearly, our ”polynomial” τ ∈ Z[Ξ] satisfies Assertion B of Theorem 1 (in fact, Ξ is the empty family, so that there exists no ξ∈Ξ, and thus Assertion B of Theorem 1 is vacuously true). Hence, it also satisfies Assertion A of Theorem 1 (because Theorem 1 states that assertionsA and B are equivalent). In other words, there exists a family (τ_d)_d∈

N|m ∈ (Z[Ξ])^N|m such that τ = w_m (τ_d)_d∈

N|m

. Since Z[Ξ] = Z, this yields the assertion of Corollary 3. Thus, Corollary 3 is proven.

References

[1] Michiel Hazewinkel, Witt vectors. Part 1, revised version: 20 April 2008.

[2] Darij Grinberg, Witt#2: Polynomials that can be written as wn. http://www.cip.ifi.lmu.de/~grinberg/algebra/witt2.pdf

[3] Darij Grinberg, Witt#3: Ghost component computations.

http://www.cip.ifi.lmu.de/~grinberg/algebra/witt3.pdf [4] Darij Grinberg, Witt#5: Around the integrality criterion 9.93.

http://www.cip.ifi.lmu.de/~grinberg/algebra/witt5.pdf