Number Theory: An Introduction to Mathematics

280 VI Hensel’sp-adic Numbers

Conversely, suppose|b− 1 | 2 ≤ 2 −^3. In Proposition 15 takeF = Q 2 and

f(x)=x^2 −b. The hypotheses of the proposition are satisfied, since|f( 1 )| 2 ≤ 2 −^3

and|f 1 ( 1 )| 2 = 2 −^1 , and henceb=a^2 for somea∈Q 2. 

Corollary 17Let b be an integer not divisible by the prime p.

If p= 2 ,thenb=a^2 for some a∈Qpif and only if b is a quadratic residue

modp.

If p= 2 ,thenb=a^2 for some a∈Q 2 if and only if b≡1mod8.

It follows from Corollary 17 thatQpcannot be given the structure of anordered

field. For, ifpis odd, then 1−p=a^2 for somea∈Qpand hence

a^2 + 1 +···+ 1 = 0 ,

where there arep−1 1’s. Similarly, ifp=2, then 1− 23 =a^2 for somea∈Q 2 and

the same relation holds with 7 1’s.

Suppose again thatFis a field with a complete non-archimedean absolute value||.

LetRandMbe the corresponding valuation ring and valuation ideal, and letk=R/M

be the residue field. For anya∈ Rwe will denote bya ̄the corresponding element

a+Mofk, and for any polynomial

f(x)=cnxn+cn− 1 xn−^1 +···+c 0

with coefficientsc 0 ,...,cn∈R, we will denote by

f ̄(x)= ̄cnxn+ ̄cn− 1 xn−^1 +···+ ̄c 0

the polynomial whose coefficients are the corresponding elements ofk.

The hypotheses of Proposition 15 are certainly satisfied if|f(a 0 )|< 1 =|f 1 (a 0 )|.

In this case Proposition 15 says that if

f ̄(x)=(x− ̄a 0 )h ̄ 0 (x),

wherea 0 ∈R,h 0 (x)∈R[x]andh 0 (a 0 )/∈M,then

f(x)=(x−a)h(x),

wherea−a 0 ∈M,andh(x)∈R[x]. In other words, the factorization off ̄(x)ink[x]
can be ‘lifted’ to a factorization off(x)inR[x]. This form of Hensel’s lemma can
be generalized to factorizations where neither factor is linear, and the result is again
known as Hensel’s lemma!
Proposition 18Let F be a field with a complete non-archimedean absolute value||.
Let R and M be the valuation ring and valuation ideal of F , and k=R/M the residue
field.
Let f ∈R[x]be a polynomial with coefficients in R and suppose there exist rela-
tively prime polynomialsφ,ψ∈k[x], withφmonic and∂(φ)> 0 , such thatf ̄=φψ.
Then there exist polynomials g,h∈R[x], with g monic and∂(g)=∂(φ),such
thatg ̄=φ,h ̄=ψand f=gh.