Totally Ramified Extensions

AMA CSC-AI

4. Structure of p-adic Fields

4.2. Totally Ramified Extensions

CD. 45. A

'L7

7L'

4. Structure of p-adic Fields 101

This is a canonical example of a ramified covering of degree e at the origin, in a topological sense: The inverse image of any z 0 consists of e distinct preimages, while u = 0 is the only preimage of z = 0. If = En>m anzn (0 < Izl < s) is as before a meromorphic function in a neighborhood of the origin, we can make the change of variable z = ue and obtain a new expansion

17(u) = (ue)

=

amen.

n>m

In this way, the field L is embedded in the field L' consisting of convergent Laurent series in the variable u. There is no function defined in a neighborhood of z = 0 in C, so that the field L' = L(zlle) is a proper extension of the field of convergent Laurent series L in the variable z. This extension L' is totally ramified over L, with degree e: It is obviously comparable to the extension Qp(7r) of Qp if -r = pile Observe that with meromorphic functions it is traditional to work with the order-of- vanishing function in, instead of a corresponding ultrametric absolute value Il; to = Bm (for a choice 0 < 0 = Izlo < 1; there is no canonical choice for 0 here).

The rational field Q can similarly be compared to the field of rational functions C(z), the completions Qp (letting now the prime p vary) corresponding to the fields of meromorphic functions near a variable point a E C instead of the origin.

"'y 1-,

F"'

and h = co is a constant. Considering that brc, = 1, the only possibility now is m = 0 and a trivial factorization.

The preceding argument mod p can be made directly on the coefficients. Let r > 1 be the smallest power of X in h having a coefficient not divisible by p:

p does not divide Cr but p divides cr_t, cr_2, ... , c0.

The coefficient of Xr in the product of g and h is

ar = bocr +b1cr_t +b2cr_2+-- = bocr + P( .

Since bOCr is not divisible by p, the preceding equality shows that p does not divide ar either. By assumption, this shows that r = n. Summing up,

n=m+-e>m>r=n

implies m = n and f = 0. The factorization g - h off is necessarily trivial.

The same proof shows the following more general result.

Let A be a factorial ring with fraction field K, and it a prime of A. Any polynomial

f (ofdegreen> 1)

with an not divisible by n, ai divisible by 7r for 0 < i < n - 1, ao not divisible by 7r2, is irreducible in the rings K[X] and A[X].

Definition. A monic polynomial f (X) E Z p [X] of degree n > 1 satisfying the conditions of the theorem, namely

f (X) = X" mod p, f (0) 0 mod p2, is called an Eisenstein polynomial

Theorem 2. Let K be a finite, totally ramified extension of Qp. Then K is generated by a root of an Eisenstein polynomial.

PROOF. The maximal ideal P of the subring R = B<t of K is principal and generated by an element 7r with I7r Ie = Ipl. Since n = [K : Qp] = e by assumption, the linearly independent powers (7ri)0<i<e generate K and K = Qp[7r]. The irreducible polynomial of this element can be factored (in a Galois extension of Qp containing K) as

f(X)=fl(X-7ra)=Xe+ E aiX`±fl7r°.

0 O<i<e ^Cr

The constant term has absolute value I IIQ7r° I = In le = IpI (by (3.3) all automor- phisms a are isometric), whereas the intermediate coefficients ai satisfy Jai I < 1

I'D

4. Structure of p-adic Fields 103

(each is divisible by one 7r° at least, and a; E Zp). Hence these intermediate coefficients are in pZp as required: f is an Eisenstein polynomial.

Examples. (1) In the field Q2, -I is not a square (I.6.6), and we can construct the quadratic extension K = Q2(i) = Q2[Xj/(X2 + 1). Since

(i+l)2=i2+2i+I=2i,

the element i + 1 is a square root of 2i. With the (unique) extension of the 2-adic absolute value we have

Ii+112=12i1=121=1,²

Ii+II=

^z,

so that i + I is a generator of the maximal ideal P of the maximal subring R of the field K: P = (i + 1)R. The quadratic extension K is totally ramified of index e = 2, hence wildly ramified. Let x = i + 1. Then x - 1 = i and (x - 1)2 = -1 shows that x is a root of the polynomial

X2-2X+2=(X- 1)2+1.

This is an Eisenstein polynomial (relative to the prime 2), and K = Q2(i) is also obtained as a splitting field of this Eisenstein polynomial.

(2) For p # 2 let us add a primitive pth root of unity to Qp. In other words, we are adding to Qp a root of p - 1 = 0 with i; # 1. Hence is a root of

cp(X)=(Xp-1)/(X - 1)=Xp-1+---+X+l.

This is the pth cyclotomic polynomial: It is irreducible, since the change of variable X - I = Y produces

(Dp(X) = [(Y+ 1)p - l]/Y = ^Yp-1 +p(...)+ p,

an Eisenstein polynomial. Hence we obtain an extension of Qp of degree p - 1 prime to p. We shall prove that it is totally ramified. Since the powers ` are also roots of the same equation when i is not a multiple of p, the powers ' (1 < i <

p - 1) form a complete set of conjugates of , and

cp(X) = fl (X -

1<<p-1 Obviously, Op(I) = 1 + - - + I = p, so that

p = (Dp(1) _ F1 (1 -

I<i<p_1

But all absolute values I 1 - ` 1 are equal by (3.3), since these elements are

°-'

.»;

inn

IIIIII

one

conjugate. The preceding inequality leads to

IpI=

1a<p-1

This proves that n = 1 - is a generator of P in R: The extension K = is ramified with degree n = e = p - 1. hence totally and tamely ramified.

In the course of the preceding deduction we have used the uniqueness of extension of valuations again. However, in the present context, it is obvious that

1 - ' = (1 - 0(1 + ... + `-1)

implies I 1 - ' I < I 1 - I. But the roles of and ` may be reversed: ' is also a generator of the cyclic group µp of order p when 1 < i < p - 1, so that is a power )i of ` (take j such that i j - 1 mod p). This furnishes the equality I 1 - ` I = I 1 - I. By the way, this proves that