A completion turns arithmetic “near one prime” into its own complete number system. Instead of asking how an algebraic number behaves simultaneously at every rational prime, we choose one prime ideal and magnify only that direction. The result is a local field.
For Q the local field at p is Q_p. For a number field K and a prime ideal P above p, the local field is the completion K_P. The global factorisation data
pO_K=∏P_i^(e_i)
then becomes a family of local extensions K_(P_i)/Q_p with ramification index e_i and residue degree f_i.
This is Guide 39 in the Bukit Timah Tutor Computational Number Theory series. It continues directly from Guide 38: Prime-Ideal Decomposition.
1. The p-adic valuation
For nonzero rational x, write
x=p^k a/b
with a,b integers not divisible by p.
Define
v_p(x)=k.
Set v_p(0)=+∞.
This valuation measures divisibility by p rather than ordinary magnitude.
2. The p-adic absolute value
Define
|x|_p=p^(−v_p(x)).
Then highly p-divisible numbers are small.
For example, in Q_5:
|25|_5=5^(−2)=1/25, |1/5|_5=5.
This reverses ordinary size intuition.
3. Ultrametric triangle inequality
The p-adic absolute value satisfies
|x+y|_p ≤ max(|x|_p,|y|_p).
This is stronger than the ordinary triangle inequality.
If |x|_p≠|y|_p, then in fact
|x+y|_p=max(|x|_p,|y|_p).
The larger p-adic term dominates; cancellation can only make a sum smaller when the two terms have equal size.
4. Completion of Q
Complete Q with respect to |·|_p. The result is Q_p.
A p-adic number can be written formally as
a_m p^m+a_(m+1)p^(m+1)+a_(m+2)p^(m+2)+...
with digits a_i in {0,…,p−1}.
The expansion extends infinitely toward higher powers of p, not toward smaller real magnitudes.
5. Example: −1 in Z_5
The identity
−1=4+4·5+4·5²+4·5³+...
is valid 5-adically.
The finite partial sum through 5^n equals
5^(n+1)−1.
Its difference from −1 is 5^(n+1), whose 5-adic absolute value tends to zero.
6. p-adic integers
The valuation ring is
Z_p={x in Q_p : v_p(x)≥0}.
Its unique maximal ideal is
pZ_p={x:v_p(x)≥1}.
The residue field is
Z_p/pZ_p ≅ F_p.
Thus Q_p has the same residue prime p but infinitely more precision above it.
7. Units and uniformisers
An element u in Z_p is a unit exactly when
v_p(u)=0.
The element p has valuation one and is a uniformiser.
Every nonzero x in Q_p can be written uniquely as
x=p^k u
with k in Z and u a p-adic unit.
8. Hensel’s lemma
Hensel lifting upgrades approximate modular roots into exact p-adic roots.
A common simple-root version says: if
f(a)≡0 (mod p) and f'(a) not≡0 (mod p),
then there is a unique α in Z_p with
α≡a (mod p) and f(α)=0.
Guide11 introduced prime-power lifting; the local-field viewpoint explains the limiting object those lifts converge to.
9. Example: a 5-adic square root
Consider
f(x)=x²−6.
Modulo5,
f(1)=−5≡0, f'(1)=2 not≡0.
Therefore the root x≡1 mod5 lifts uniquely to a square root of6 in Z_5.
The root x≡−1 mod5 gives the other square root.
10. Finite extensions of Q_p
Let L/Q_p be a finite extension.
There is a unique extension of the p-adic absolute value to L.
The valuation ring is
O_L={x in L:v_L(x)≥0}
with maximal ideal
P_L={x:v_L(x)>0}.
The residue field is finite:
O_L/P_L ≅ F_(p^f).
11. Ramification index and residue degree
Choose the integer normalisation v_L(π)=1 for a local uniformiser π.
Then
v_L(p)=e.
The residue degree is
f=[O_L/P_L:F_p].
For finite extensions of p-adic fields,
[L:Q_p]=ef.
This is the local version of the global degree relation from Guide38.
12. Unramified extensions
An extension L/Q_p is unramified when
e=1.
Then
[L:Q_p]=f
and all degree growth appears in the residue field.
For every f≥1 there is, up to Q_p-isomorphism, a unique unramified extension of degree f.
13. Constructing an unramified extension
Choose a monic polynomial whose reduction modulo p is irreducible and separable of degree f.
Lift it to Z_p[x] and adjoin a root.
For example over Q_5,
g(x)=x²+x+1
has no root in F_5, so it is irreducible modulo5.
Adjoining a root gives an unramified quadratic extension with
e=1, f=2.
14. Eisenstein extensions
A monic polynomial
x^n+a_(n−1)x^(n−1)+...+a0
over Z_p is Eisenstein when p divides every nonleading coefficient, p² does not divide a0, and the leading coefficient is a unit.
An Eisenstein polynomial is irreducible over Q_p and adjoining a root produces a totally ramified extension of degree n.
15. Worked ramified extension Q_5(√5)
The polynomial
x²−5
is Eisenstein at5.
Therefore
L=Q_5(√5)
has
[L:Q_5]=2, e=2, f=1.
The element
π=√5
is a local uniformiser and
π²=5.
With v_L(π)=1, we have v_L(5)=2.
16. Global-to-local completion
Let K be a number field and P a prime ideal of O_K above p.
Complete K with respect to the P-adic absolute value to obtain
K_P.
Then
[K_P:Q_p]=e(P/p)f(P/p).
Each prime ideal above p produces one local factor.
17. Tensor-product decomposition
There is a canonical decomposition
K⊗_Q Q_p ≅ ∏_(P|p) K_P.
This formula packages every completion above p at once.
A split prime produces several smaller local factors; an inert prime produces one larger unramified factor; a ramified prime produces a factor where the valuation scale changes.
18. Return to K=Q(√5)
Use the prime decompositions from Guide38.
At p=11:
11O_K=P4P8.
Each prime has e=f=1.
Therefore
K⊗Q_11 ≅ Q_11 × Q_11.
Both completions are degree-one copies of Q_11.
19. The inert prime p=3
For p=3,
3O_K=P
with
e=1, f=2.
Therefore
K_P/Q_3
is an unramified quadratic extension.
The residue field is F_9.
20. The ramified prime p=5
For p=5,
5O_K=P²
with
e=2, f=1.
Therefore the completion is ramified quadratic:
K_P ≅ Q_5(√5).
The residue field remains F_5 while the valuation step becomes finer.
21. Why local fields simplify global equations
Suppose a Diophantine equation has a rational or algebraic integer solution globally.
Then it must have a solution in every completion:
R, Q_p, or K_P
as appropriate.
Therefore local impossibility is a certificate of global impossibility.
This produces powerful congruence and valuation obstructions before expensive global searches begin.
22. Local solvability is not always global solvability
The converse can fail: an equation may have solutions in every local completion and still have no global solution.
This local–global gap leads to the Hasse principle, Selmer groups and Brauer–Manin obstructions.
Computationally, local testing remains invaluable even when it is not sufficient.
23. Local norms
For L/K a finite local-field extension, the norm map
N_(L/K):L*→K*
interacts strongly with valuations:
v_K(N(x)) = f·v_L(x)
under the integer valuation normalisations v_L(π_L)=1 and v_K(π_K)=1.
Unit norms contain subtler residue and ramification information.
Local norm tests become building blocks for global norm equations in Guide40.
24. Tame and wild ramification
Let the residue characteristic be p.
Ramification is tame when
p does not divide e.
It is wild when
p divides e.
Wild ramification produces deeper higher-ramification phenomena and usually more difficult discriminant behaviour.
25. Different and discriminant
The different ideal measures failure of the trace-dual lattice to equal the local ring of integers.
Its norm gives the relative discriminant.
Locally, the exponent of the different records ramification intensity. Unramified extensions have trivial different; ramified extensions have positive different exponent.
26. Hensel factorisation
Hensel’s lemma lifts not only roots but coprime polynomial factorisations.
If
f_bar=g_bar h_bar mod p
with gcd(g_bar,h_bar)=1, then the factorisation lifts uniquely to compatible factors over Z_p to arbitrary precision.
This turns finite-field factorisation into p-adic factorisation.
27. Precision is part of the data structure
Computer p-adic arithmetic is usually finite precision.
An element may be stored as
a + O(p^N).
This means the value is known modulo p^N, not that the omitted terms are zero.
Algorithms must track whether additions, multiplications, divisions and root lifting preserve enough precision for the claimed output.
28. Relative versus absolute precision
For x with valuation k, absolute precision N describes knowledge modulo p^N.
Relative precision is roughly N−k significant p-adic digits beyond the leading valuation.
Dividing by a highly p-divisible element can lose relative precision, just as subtracting nearly equal real numbers can lose floating-point significance.
29. Local fields in computational number theory
Local fields appear in:
Hensel lifting prime-ideal factorisation local norm tests p-adic root finding p-adic regulators class-field computations elliptic-curve local data Galois representations Diophantine sieves.
They are the natural environment whenever one prime dominates the arithmetic question.
30. SageMath local-field models
SageMath provides Qp/Zp base fields, unramified extensions and Eisenstein extensions. Its p-adic tutorial explicitly distinguishes unramified extensions constructed by residue-field degree from Eisenstein extensions producing ramification.
Software objects also expose ramification index and residue/inertia degree.
A calculation should record both the defining polynomial and the precision model.
31. Verification receipt
For a local extension L/Q_p, retain:
defining polynomial irreducibility evidence uniformiser ramification index e residue degree f check ef=[L:Q_p] residue-field polynomial precision cap / model.
For a global completion K_P, also retain the global prime ideal P and its e,f data.
32. Common mistakes
1. Confusing real size with p-adic size. 2. Forgetting that completion depends on a chosen prime. 3. Calling every irreducible p-adic extension unramified. 4. Treating an Eisenstein extension as residue-degree growth.
5. Forgetting [L:Q_p]=ef. 6. Ignoring finite precision in computer arithmetic. 7. Assuming local solvability implies global solvability. 8. Mixing the normalisation v_L(π)=1 with a valuation normalised to extend v_p without stating the convention.
33. Practice set
1. Define v_p(x). 2. Compute |25|_5. 3. State the ultrametric inequality. 4. What is the residue field of Z_p?
5. State simple-root Hensel lifting. 6. Why is x²+x+1 an unramified quadratic generator over Q_5? 7. Why is x²−5 totally ramified? 8. Give e,f for Q_5(√5).
9. Describe the Q_11 completions of Q(√5). 10. Describe the Q_3 completion. 11. Describe the Q_5 completion. 12. Distinguish tame from wild ramification.
34. Answers
1. The exponent of p in x. 2. 1/25. 3. |x+y|≤max(|x|,|y|). 4. F_p.
5. A simple root mod p lifts uniquely to Z_p. 6. Its reduction is irreducible degree2 mod5, giving e=1,f=2. 7. It is Eisenstein at5. 8. e=2,f=1.
9. Two degree-one completions, each Q_11. 10. One unramified quadratic extension of Q_3. 11. Q_5(√5), ramified quadratic. 12. Tame means residue characteristic does not divide e; wild means it does.
Sources and further study
SageMath’s p-adics reference documents p-adic fields and extensions. Its p-adic tutorial gives explicit unramified and Eisenstein extension constructions, while the valuation reference treats ramification tests on number fields and completions.
Continue through Batch 10
Return to Guide 38. Continue to Guide 40: S-Units, Norm Equations, Thue Equations and Lattice Reduction.