ON THE RING OF NEWTON INTERPOLATING SERIES OVER A LOCAL FIELD

ON THE RING OF NEWTON INTERPOLATING SERIES OVER A LOCAL FIELD

GHIOCEL GROZA and AZEEM HAIDER

Newton interpolating polynomials at m distinct nodes x0, x1, . . . , xm−1 of mul- tiplicities ni, i= 0,1, . . . , m−1,have been used to prove the transcendence of the value of the exponential function at an algebraic number [6]. In this paper, by using these polynomials, we construct a Banach algebra over a local non- archimedean field K which contains the Tate algebra in n indeterminates over K. Many properties of this Banach algebra are analogues of those of the Tate algebras.

AMS 2000 Subject Classification: 13J05, 12J25, 12J27, 32P05, 46J05.

Key words: formal power series, noetherian ring, Tate algebra.

1. INTRODUCTION

The Tate algebrasT_n over a complete non-archimedean fieldK,consist- ing of all power series innvariables converging on the “closed” unit polydisc in Kⁿ, lead to the definition of affinoid algebras ([2], Part B or [3], Ch. 2).

Motivated by the question how to characterize elliptic curves with bad reduc- tion this category of analytic-algebraic objects was born in 1961 when J. Tate gave a seminar at Harvard entitled “Rigid Analytic Spaces”. This category of objects make possible analytic continuation over totally disconnected ground fields. A generalization of Tate algebras by using power series is given in [5].

In this paper, we consider a commutativeK-algebraRover an arbitrary fieldK and distinct elementsx₀, x₁, . . . , x_m−1 ofK.In Section 2 we construct a commutative R-algebra Rx₀,...,x_m−1[[X]] of formal series in nindeterminates which is noetherian whenRis noetherian (see Theorem 2.3). Then we describe the units of this algebra which in the case m = 1 and x₀ = 0 becomes the R-algebra of formal power series. In Section 3, we consider a local fieldR=K and define aK-subalgebraT K_{x0,...,xm−1}[[X]] ofK_{x0,...,xm−1}[[X]] including Tate algebra Tn which contains series converging in the “closed” unit polydisk.

The corresponding residue algebra is isomorphic to K(^x0,...,x_m−1)[[X]], where K is the residue field of K(see Proposition 3.2) and each series gives rise to

MATH. REPORTS9(59),4 (2007), 343–356

a continuous function. We describe the units of T Kx₀,...,x_m−1[[X]] and prove that it is a noetherian domain (see Theorem 3.6).

2. THE ALGEBRA OF NEWTON INTERPOLATING SERIES

Let K be a commutative field, m ≥ 1 an integer and x₀, x₁, . . . , xm−1

distinct elements of K.Ifn is a nonnegative integer we consider the quotient q(n) and the remainderr(n) obtained whenn is divided bym.Construct the polynomials

(1) un=

m−1 k=0

(X−xk)^{q(n)+σ(r(n)−k−1)}∈K[X], n= 0,1,2, . . . ,

where σ(x) is equal to 0 if x < 0 and 1 otherwise. Thus, u₀ = 1, u₁ = X−x₀, . . . , um = (X−x₀). . .(X−xm−1) and

(2) u_i+jm=u_iu^j_m.

IfP ∈K[X],we denote by P_t(z₁;z₂;. . .;z_t) the divided difference of P with respect tot distinct elements z₁, z₂, . . . , zt∈K. Thus,P₁(z₁) =P(z₁) and (3) Ps(z₁;z₂;. . .;zs) = Ps−1(z₂;. . .;zs)−Ps−1(z₁;. . .;zs−1)

z_s−z₁ , for 2≤s ≤t. Letd < tbe the degree ofP. Then, for everyx∈K,by Newton interpolating formula, we have

(4) P(x) =

d+1

s=1

P_s(z₁;z₂;. . .;z_s)u_s−1(x).

Lemma 2.1. Let K be a commutative field and x₀, x₁, . . . , xm−1 dis- tinct elements of K. Then there exist elements c_k(s, t) ∈ K, where s, t ∈ {0,1, . . . , m−1}, max{s, t} ≤ k ≤ s+t, such that in the ring K[X], for all nonnegative integers iand j,

(5) u_iu_j =

r(i)+r(j)

k=max{r(i),r(j)}

c_k(r(i), r(j))u_{(q(i)+q(j))m+k}, where c_k(s, t) = u_{min{s,t};k−max{s,t}+1}

x_max{s,t};x_max{s,t}+1;. . .;x_k

(the di- vided difference ofu_min{s,t} with respect tox_max{s,t}+1, . . . , xk)and xk=x_r(k) ifk≥m.

Proof. If r(j) = 0 then j = q(j)m and, by (2), u_iu_j =u_i+j. Hence (5) follows withct(t,0) = 1. Now, suppose that r(j) = 0.Then

(6) u_iu_j =u_(q(i)+q(j))mu_r(i)u_r(j).

Assume r(i) ≥ r(j). Then, by Newton interpolation formula with respect to x_r(i);x_r(i)+1;. . .;x_r(i)+r(j),we have

(7) u_r(j)=

r(j)

v=0

u_r(j);v+1

x_r(i);x_r(i)+1;. . .;x_r(i)+v ^v

w=1

X−x_r(i)+w−1 . Now, (6) and (7) imply (5).

LetRbe a commutativeK-algebra andx₀, x₁, . . . , x_m−1distinct elements of K. If ν = (ν₁, ν₂, . . . , νn) ∈ Nⁿ, set N(ν) = ν₁ +ν₂ +· · · +νn, X = (X₁, . . . , Xn), Uν =

n

j=1uν_j ∈ K[X], where uν_j ∈ K[Xj] are given by (1).

Putq(ν) = (q(ν₁), q(ν₂), . . . , q(ν_n)), r(ν) = (r(ν₁), r(ν₂), . . . , r(ν_n)) and if τ = (τ₁, τ₂, . . . , τ_n) ∈Nⁿ, j∈N,defineν+τ = (ν₁+τ₁, ν₂+τ₂, . . . , ν_n+τ_n) and jν = (jν₁, jν₂, . . . , jνn). Order Nⁿ in the following manner: ν < τ if either N(ν) < N(τ) or N(ν) = N(τ) and ν is less than τ with respect to the lexicographical order. Denote by ∞ⁿ a symbol such that ν < ∞ⁿ for every ν ∈ Nⁿ and by R_{(x0,...,xm−1)} [[X]] the set of formal series of the form f = ^∞

n

ν=0a_νU_ν, a_ν ∈ R. An element of R_{(x0,...,xm−1)} [[X]] is called a Newton interpolating seriesat x₀, x₁, . . . , x_m−1.Iff, g= ^∞

n

ν=0b_νU_ν ∈R_{(x0,...,xm−1)}[[X]], then define addition and multiplication off and g by

(8) f +g=

∞ⁿ

ν=0

(aν +bν)Uν,

(9) f g=

∞ⁿ

ν=0

pνUν, where

(10) p_ν =

max{α+q(β)m,β+q(α)m}≤ν≤α+β

C_{ν−(q(α)+q(β))m}(r(α), r(β))a_αb_β, withC_ν(α, β) =c_ν1(α₁, β₁). . . c_νn(α_n, β_n) andc_i(s, t) defined in Lemma 2.1.

Consider a non-zero series f = ^∞

n

ν=0a_νU_ν ∈ R_{(x0,...,xm−1)}[[X]]. If τ is the smallest subscript ν for which a_ν is different from zero, then τ will be calledthe order of f and will be denoted o(f).We agree to attach the order

∞ⁿ to the element 0 of R_{(x0,...,xm−1)}[[X]]. It follows in the usual way that o(f+g) ≥min{o(f), o(g)},but o(f g)≥o(f) +o(g) does not hold for every f, g∈R_{(x0,...,xm−1)}[[X]].

Lemma2.2. There exists an injective map ϕ:R[X]→R_{(x0,...,xm−1)}[[X]]

such that

ϕ(P +Q) =ϕ(P) +ϕ(Q) for allP, Q∈R[X],and

ϕ(P Q) = ϕ(P) ϕ(Q),

where the addition and the multiplication in R_{(x0,...,xm−1)}[[X]] are defined by (8)and(9).

Proof. For P ∈R[Xj], j ∈ {1,2, . . . , n} with d= deg(P) ≤ m−1 , by means of Newton interpolation formula, put

(11) ϕ(P) =

m−1

k=0

P_k+1(x₀;x₁;. . .;x_k)(X_j−x₀). . .(X_j−x_k−1).

Ifd≥mthen there existQ_s ∈R[X_j], s= 0, . . . ,_d

m

, with deg(Q_s)< m, such

thatP = [_m^d]

s=0Qsu^s_m.Then defineϕ(P) = [_m^d]

s=0ϕ(Qs)u^s_m. Now, ifP =

s

i=1aiX_tⁱ ∈R[X₁, . . . , Xt], t≤n,whereai ∈R[X₁, . . . , Xt−1], define

ϕ(P) = s i=1

ϕ(a_i)ϕ(X_t)ⁱ

by induction on t. Because a system of polynomials which have different degrees is linearly independent over K, the proof is complete.

Theorem2.3. If R is a commutative K-algebra which is a noetherian ring, then under the addition and the multiplication defined by (8) and (9), the setR_{(x0,...,xm−1)}[[X]] becomes a commutativeK-algebra which is a noethe- rian ring.

Proof. Since for every nonnegative integerssandtthe system of polyno- mialsu_(q(s)+q(t))m(X), u_{(q(s)+q(t))m+1}(X), . . . , us+t(X)∈R[X] is linearly inde- pendent overK, by (5), for each i, p_i(s, t) =p_i(t, s).If we also denoteϕ(R[X]) byR[X],where ϕis given in Lemma 2, we obtain that R[X] becomes a com- mutative K-algebra when the addition and the multiplication are defined by (8) and (9). Consider f = ^∞

n

ν=0aνUν ∈R_{(x0,...,xm−1)}[[X]], and take the partial sumfs=

s

ν=0aνUν.Then each coefficient off+g, f gfrom (8) and (9) can be obtained as a sum or a multiplication of polynomials of the form fs and gs.

Hence the statement that R_{(x0,...,xm−1)}[[X]] is a commutative K-algebra can be reduced toR[X], whenceR_{(x0,...,xm−1)}[[X]] is a commutativeK-algebra.

Since theK-algebra R_{(x0,...,xm−1)}[[X₁, . . . , X_n−1]]_{(x0,...,xm−1)}[[X_n]] is iso- morphic to R_{(x0,...,xm−1)}[[X₁, . . . , Xn]], we prove by induction on n that the K-algebra R_{(x0,...,xm−1)}[[X]] is a noetherian ring. It is enough to show that R_{(x0,...,xm−1)}[[X]] is a noetherian ring. Let I be an ideal in R_{(x0,...,xm−1)}[[X]].

In a similar way to the particular casem= 1, x₀ = 0 (see [7], p. 138), for any integeri≥0 denote byE_i(I) the set of elements ofR consisting of 0 and the coefficients ofui in the elements of I of orderi. Then Ei(I) is an ideal of R.

Since u_iu_m =u_i+m,forr ∈ {0,1, . . . , m−1}, the non-zero ideals E_r+jm(I), j∈N,constitute an ascending sequence. Thus there existsjr∈Nsuch that

Er+j_rm(I) =

j∈N

Er+jm(I), r∈ {0,1, . . . , m−1}.

Hence there existb₁,r, . . . , bn_r,r ∈Er+j_rm(I) such thatEr+j_rm(I) is generated by these elements. Considerg_1,r, . . . , g_nr,r∈ I such thato(g_i,r) =r+j_rm for r∈ {0,1, . . . , m−1},and

(12) gi,r=bi,ru_o(gi,r)+ ∞ j>o(g_i,r)

di,r,juj, i∈ {0,1, . . . , nr}.

Denote bys the greatest integer among the order of the series gi,r.Now, for every t < s consider c_1,t, . . . , c_mt,t ∈ E_t(I) such that E_t(I) is generated by these elements and chooseh₁,t, . . . , hm_t,t∈ I such that, for t < s,

(13) h_i,t=c_i,tu_o(hi,t)+ ∞ j>o(h_i,t)

e_i,t,ju_j, i∈ {0,1, . . . , m_t}.

We shall prove that the ideal I is generated by the elements gi,r, hi,t

given by (12) and (13). Let f = ^∞

i=o(f)

a_iu_i be an element of I. If o(f) < s thena_o(f)∈ E_o(f)(I) and we can write

(14) a_o(f)=

m_o(f)

j=1

α_jc_j,o(f),

where α_j ∈ R. Hence the order of f −^mo(f)

j=1 α_jh_j,o(f) is greater than o(f). It follows by successive applications of this result that we can considero(f)≥s.

Then we can write

(15) a_o(f)=

n_r

j=1

βjbj,r,

whereβj ∈R and r=r(o(f)).Hence the order of f −

n_o(f)

j=1 βju^qm^(o(f))−jrgj,r is greater thano(f).By successive applications of this result, for everyn we get elementsvj,r,n ∈R_{(x0,...,xm−1)}[[X]] such that the order off−^m−1

r=0 n_r j=1

n

i=1vj,r,igj,r

is greater thann ando(v_j,r,i) tends to infinity withifor fixedjand r. Hence vj,r= ^∞

i=1vj,r,i∈R_{(x0,...,xm−1)}[[X]] andf =

m−1 r=0

nr

j=1vj,rgj,rbelongs to the ideal generated by the elementsg_j,r, h_j,t.

Theorem 2.4. If R is a commutative K-algebra and f = ^∞

i=0aiui ∈ R_{(x0,...,xm−1)}[[X]],then f is a unit in R_{(x0,...,xm−1)}[[X]] if and only if

(16) dj(f) =

j i=0

cj(i, j)ai

are units inR for everyj∈ {0,1, . . . , m−1}.

Proof. Since f is a unit in R_{(x0,...,xm−1)}[[X]] if and only if there exists g = ^∞

i=0b_iu_i ∈ R_{(x0,...,xm−1)}[[X]] such that p₀ = 1 and p_i = 0, for all i with i≥1,wherep_i are given by (10), we have

(17)

max{s+q(t)m,t+q(s)m}≤i≤s+t

c_{i−(q(s)+q(t))m}(r(s), r(t))a_sb_t=δ_i,0,

whereδi,j is the Kronecker delta symbol. But, for a fixedi,the corresponding equation of (17) contains elements b_j with j ≤ i and the coefficient of b_i is d_r(i) given by (16). Then, if i₁≡i₂ (modm),the coefficients ofbi₁ and bi₂ in the corresponding equations to (17) fori=i₁ andi=i₂, respectively, are the same. Hence, if d_j(f) are units inR for every j∈ {0,1, . . . , m−1},thenf is a unit inR_{(x0,...,xm−1)}[[X]].

Now, let f be a unit in R_{(x0,...,xm−1)}[[X]]. Then a₀b₀ = 1 and d₀(f) = c₀(0,0)a₀ =a₀ is a unit inR. We prove by induction onj thatdj(f) is a unit inRfor everyj∈ {0,1, . . . , m−1}. Sinced₀(f) is a unit, we can suppose that dj(f) anddj(g) =

j

i=0cj(i, j)bi are units in R forj < h≤m−1, and we have to prove thatd_h(f) also is a unit inR. LetI be the ideal generated byd_h(f)

inR. Add to equation (17) fori= 1 multiplied by c_h(1, h−1),the equation fori= 2 multiplied by c_h(2, h−1) and so on, finally, the equation for i=h multiplied byc_h(h, h−1).We get

c_h(1, h−1) 1

t=0

₁

s=1−t

c₁(s, t)as

bt+c_h(2, h−1) 2 t=0

₂

s=2−t

c₂(s, t)as

bt+· · ·

(18) +c_h(h, h−1) h t=0

_h

s=h−t

c_h(s, t)a_s

b_t∈ I.

Since dh(f) =

h

i=0ch(i, h)ai = a₀ +

h

s=1ch(s, h)as ∈ I, we replace a₀ by

− ^h

s=1c_h(s, h)as.Thus, it follows by (18) that ch(1, h−1)

− h s=1

ch(s, h)asb₁+c₁(1,0)a₁b₀+c₁(1,1)a₁b₁

+

+c_h(2, h−1)

− h s=1

c_h(s, h)a_sb₂+ 1

t=0

₂

s=2−t

c₂(s, t)a_s

b_t+ 2 s=1

c₂(s,2)a_sb₂

+· · ·

+c_h(h−1, h−1)

− h s=1

c_h(s, h)a_sb_h−1+

h−2

t=0

_h−1

s=h−1−t

c_h−1(s, t)a_s

b_t+

+

h−1

s=1

c_h−1(s, h−1)a_sb_h−1

+c_h(h, h−1) h−1

t=0

_h

s=h−t

c_h(s, t)a_s

b_t∈ I, or

(19)

h−1

t=0



^h

s=1



^min{

s+t,h}

k=max{s,t}

c_h(k, h−1)c_k(s, t)−c_h(t, h−1)c_h(s, h)



a_s



b_t∈I, where c_i(j, k) = 0 if i < max{j, k} or i > j+k. But, by Lemma 2.1, the coefficiente_s,t ofa_sb_tin (19) is

(20) e_s,t =

min{s+t,h} k=max{s,t}

c_h(k, h−1)c_k(s, t)−c_h(t, h−1)c_h(s, h) =

=

min{s+t,h} k=max{s,t}

u_{min{k,h−1};h−max{k,h−1}+1}

x_max{k,h−1};. . .;xh

−

−u_{min{s,t};k−max{s,t}+1}

x_max{s,t};. . .;x_k

−u_t;2(x_h−1;x_h)u_s;1(x_h).

Hence, ifs+t=h and s≥t, then (21)

e_s,t=

h−1

k=s

u_k;2(x_h−1;x_h)u_t;k−s+1(x_s;. . .;x_k)+u_h−1;1(x_h)u_t;h−s+1(x_s;. . .;x_h)−

−ut;2(xh−1;xh)us;1(xh) = 1

x_h−x_h−1 ((us(xh)−us(xh−1))ut;1(xs)+

+ (u_s+1(x_h)−u_s+1(x_h−1))u_t;2(x_s;x_s+1) +· · ·+ (u_h−1(x_h)−u_h−1(x_h−1))·

·u_t;h−s(xs;. . .;x_h−1)) +u_h−1;1(x_h)u_t;h−s+1(xs;. . .;x_h)−ut;2(x_h−1;x_h)·

·u_s;1(x_h) = 1

x_h−x_h−1 ((u_s(x_h)−u_s(x_h−1))u_t;1(x_s) + (u_s(x_h)(x_h−x_s)−

−u_s(x_h−1)(x_h−1−x_s))u_t;2(x_s;x_s+1)+· · ·+(u_s(x_h)(x_h−x_s). . .(x_h−x_h−2)−

−us(xh−1) (xh−1−xs). . .(xh−1−xh−2))ut;h−s(xs;. . .;xh−1) + (xh−x₀). . . (x_h−x_h−2) (x_h−x_h−1)u_t;h−s+1(x_s;. . .;x_h))−u_t(x_h)−u_t(x_h−1)

xh−xh−1 u_s(x_h) =

= u_s(x_h)

xh−xh−1(u_t;1(x_s) + (x_h−x_s)u_t;2(x_s;x_s+1) +· · ·+ (x_h−x_s)·

·. . .(x_h−x_h−2)u_t;h−s(x_s;. . .;x_h−1) + (x_h−x_s). . .(x_h−x_h−1)·

·u_t;h−s+1(x_s;. . .;x_h)−u_t(x_h) +u_t(x_h−1))− u_s(x_h−1) xh−xh−1·

·(u_t;1(x_s) + (x_h−1−x_s)u_t;2(x_s;x_s+1) +· · ·+ (x_h−1−x_s). . .(x_h−1−x_h−2)·

· u_t;h−s(xs;. . .;x_h−1)) = us(x_h)

x_h−x_h−1 (ut(x_h)−ut(x_h) + ut(x_h−1))−

− us(x_h−1)

x_h−x_h−1ut(x_h−1) =ut(x_h−1)us;2(x_h−1;x_h) =c_h−1(t, h−1)c_h(s, h−1), and similarly fors+t=h ands < t.

Now, supposes+t < hand s≥t. Then e_s,t=

s+t

k=s

u_k;2(x_h−1;x_h)u_t;k−s+1(x_s;. . .;x_k)−u_t;2(x_h−1;x_h)u_s;1(x_h) =

= 1

xh−xh−1 ((u_s(x_h)−u_s(x_h−1))u_t;1(x_s) + (u_s+1(x_h)−u_s+1(x_h−1))·

·ut;2(xs;xs+1) +· · ·+ (us+t(xh)−us+t(xh−1))ut;t+1(xs;. . .;xs+t))−

−u_t;2(x_h−1;x_h)u_s;1(x_h) = 1

xh−xh−1((u_s(x_h)−u_s(x_h−1))u_t:1(x_s)+

+ (u_s(x_h)(x_h−x_s)− u_s(x_h−1) (x_h−1−x_s))u_t;2(x_s;x_s+1) +· · ·+ (u_s(x_h)·

·(x_h−xs). . .(x_h−xs+t−1)−us(x_h−1) (x_h−1−xs). . .(x_h−1−xs+t−1))·

·u_t;t+1(x_s;. . .;x_s+t)−(u_t(x_h)−u_t(x_h−1))u_s(x_h)) = u_s(x_h)

x_h−x_h−1 (u_t;1(x_s)+

+(x_h−x_s)u_t;2(x_s;x_s+1)+. . .+ (x_h−x_s). . .(x_h−x_s+t−1)·u_t;t+1(x_s;. . .;x_s+t)−

−ut(xh) +ut(xh−1))− us(xh−1)

x_h−x_h−1(ut;1(xs) + (xh−1−xs)·

·u_t;2(x_s;x_s+1) +· · ·+ (x_h−1−x_s). . .(x_h−1−x_s+t−1)·u_t;t+1(x_s;. . .;x_s+t)) =

= u_s(x_h)

x_h−x_h−1 (u_t(x_h)−u_t(x_h) +u_t(x_h−1))− u_s(x_h−1)

x_h−x_h−1(u_t(x_h−1)) =

=u_t(x_h−1)us(x_h)−us(x_h−1)

x_h−x_h−1 =u_t;1(x_h−1)u_s;2(x_h−1;x_h) and

(22) e_s,t =c_h−1(t, h−1)c_h(s, h−1).

Similarly, ifs+t < h andt≥swe also obtain (22).

Now, it follows by (19)–(22) that (23)

h−1

t=0

c_h−1(t, h−1) _h

s=1

c_h(s, h−1)a_s

b_t∈ I, or _h−1

t=0

ch−1(t, h−1)bt

_h

s=1

ch(s, h−1)as

=dh−1(g) _h

s=1

ch(s,h−1)as

∈I. Sinced_h−1(g) is a unit, we obtain

(24)

h s=1

ch(s, h−1)as∈ I. Because

h s=0

c_h(s, h)a_s, h s=1

c_h(s, h−1)a_s ∈ I, we have

c_h(h, h−1) h s=0

c_h(s, h)a_s−c_h(h, h) h s=1

c_h(s, h−1)a_s∈ I, or

(25) E =

h−1

s=0

(ch(h, h−1)ch(s, h)−ch(h, h)ch(s, h−1))as∈ I.

But, by Lemma 2.1,

(26) E =

h−1

s=0

(u_h−1;1(x_h)us;1(x_h)−u_h;1(x_h)us:2(x_h−1;x_h))as=

=

h−1

s=0

(uh−1(xh)us(xh)−uh−1(xh) (us(xh)−us(xh−1)))as=

=u_h−1(x_h)

h−1

s=0

us(x_h−1)as =u_h−1(x_h)d_h−1(f).

Now, since u_h−1(x_h) and d_h−1(f) are units in R, by (25) and (26), we obtain thatI =R and dh(f) is a unit in R.The proof is complete.

Corollary2.5. If R is a commutative K-algebra and f = ^∞

n

ν=0a_νU_ν is an element ofR_{(x0,...,xm−1)}[[X]], then f is a unit in this algebra if and only if for every ν = (ν₁, ν₂, . . . , ν_n) ∈ Nⁿ with ν_i ∈ {0,1, . . . , m−1}, 1 ≤ i ≤ n, we have

(27) d_ν(f) =

τ_i≤ν_i

C_ν(τ, v)a_τ,

where Cν(τ, v) =cν₁(τ₁, v₁)cν₂(τ₂, v₂). . . cν_n(τn, vn) are units in R.

Proof. Since the K-algebra R_{(x0,...,xm−1)}[[X₁, . . . , X_n]] is isomorphic to R_{(x0,...,xm−1)}[[X₁, . . . , Xn−1]]_{(x0,...,xm−1)}[[Xn]], the result follows by induction from Theorem 2.4.

3. A BANACH ALGEBRA OF NEWTON INTERPOLATING SERIES

Consider a field K and a non-trivial non-archimedean absolute value | | onK.Forx, y∈Kputd(x, y) =|x−y|.Then (K, d) is a metric space and we can introduce the customary topological concepts into such of space in terms of the metric. IfK is a locally compact field, then it is called a local field.

If A is a commutative ring with identity and is a non-archimedean norm on A, consider the sets see ([2], Chapter 1): A=^◦ {x∈A;x ≤1}, A∨= {x∈A;x<1} . Then A^◦ is a commutative ring with identity and A^∨ is an ideal in A .^◦ Denote the residue ring A /^◦ A^∨ by A. Let A = K be a commutative field with a non-trivial non-archimedean absolute value| |.Then K◦ is a local ring called the valuation ring of | | and K^∨ is the maximal ideal of K .^◦ If K is a local field, then it is a complete field and the residue field K is a finite field of order m = p^s, where p is the characteristic of K ([1],