A rational prime does not remain one indivisible object when it enters a number field. In the ring of integers O_K, the ideal (p) can split into several prime ideals, remain prime, or become a power of a prime ideal. The exponents record ramification; the residue-field degrees record how much finite-field extension appears at each prime above p.
This is the arithmetic microscope of algebraic number theory. Prime decomposition controls local fields, ideal norms, zeta functions, class groups, modular reduction and the exact arithmetic used in the number field sieve and CM algorithms.
This is Guide 38 in the Bukit Timah Tutor Computational Number Theory series. It assumes the maximal-order distinction from Guide 37.
1. The factorisation problem
Let K be a number field of degree n and O_K its ring of integers. For a rational prime p, factor the principal ideal
pO_K.
Unique factorisation of ideals gives
pO_K = P1^e1 ... Pg^eg
with distinct prime ideals P_i.
The integers e_i are ramification indices.
2. Residue fields
Each prime ideal P_i gives a finite residue field
O_K/P_i.
Its size is a power of p:
#(O_K/P_i)=p^(f_i).
The integer f_i is the residue degree.
Equivalently,
f_i=[O_K/P_i : F_p].
3. Fundamental degree relation
For every rational prime p,
Σ_(i=1 to g) e_i f_i = [K:Q]=n.
This is a basic consistency check for every prime-decomposition computation.
In a quadratic field, the possibilities are therefore:
split: (e,f)=(1,1),(1,1) inert: (e,f)=(1,2) ramified: (e,f)=(2,1).
4. Norm of a prime ideal
The absolute ideal norm is
N(P_i)=#(O_K/P_i)=p^(f_i).
Then
N(pO_K)=p^n
and multiplicativity gives
p^n=∏ p^(e_i f_i),
recovering the degree relation.
5. Dedekind factorisation theorem
Suppose K=Q(α), α is integral, f is the minimal polynomial of α and p does not divide the index
[O_K:Z[α]].
Factor f modulo p:
f_bar = φ1^e1 ... φg^eg
into distinct monic irreducible φ_i in F_p[x].
Then
pO_K = P1^e1 ... Pg^eg, P_i=(p, φ_i(α)), f_i=deg φ_i.
This is one of the most useful translations from polynomial factorisation to ideal factorisation.
6. Why the index condition matters
If p divides [O_K:Z[α]], factorisation of f modulo p can give misleading multiplicities or incomplete ideal information.
The problem is not Dedekind-domain factorisation itself. The problem is using coordinates from a nonmaximal order.
Therefore the safe sequence is:
compute / certify O_K choose an integral generator or basis check the p-index condition then interpret f mod p.
7. Running field: Q(√5)
Let
K=Q(√5), ω=(1+√5)/2.
From Guide37,
O_K=Z[ω]
and ω satisfies
f(x)=x²−x−1.
The field discriminant is
D_K=5.
Because O_K=Z[ω], the index is one and Dedekind factorisation applies directly at every rational prime.
8. Split example: p=11
Factor f modulo11:
x²−x−1 ≡ (x−4)(x−8) (mod11).
Therefore
11O_K=P4 P8
where
P4=(11,ω−4), P8=(11,ω−8).
Both exponents are one and both residue degrees are one.
9. Verify the roots modulo 11
Check:
4²−4−1=11≡0 (mod11), 8²−8−1=55≡0 (mod11).
The two roots are distinct, so there is no repeated factor.
The norm data is
N(P4)=11, N(P8)=11.
Hence
N(11O_K)=11²
as required for a quadratic field.
10. Residue-field maps in the split case
Modulo P4,
ω≡4.
Thus every a+bω reduces to
a+4b mod11.
The residue field is simply F_11.
Modulo P8, use ω≡8 instead.
These two reductions are distinct embeddings of the quadratic arithmetic into the same base finite field.
11. Inert example: p=3
Reduce
f(x)=x²−x−1
modulo3.
Testing x=0,1,2 gives values 2,2,1, so there is no root.
A quadratic polynomial with no root over F_3 is irreducible.
Therefore
3O_K=P
is prime.
12. Residue degree in the inert case
The residue field is
O_K/P ≅ F_3[x]/(x²−x−1),
a field with
3²=9
elements.
Thus
e=1, f=2, N(P)=9.
The rational prime remains one prime ideal but its residue field doubles in degree.
13. Ramified example: p=5
Modulo5,
x²−x−1 ≡ x²−x+4 ≡ (x−3)².
Therefore
5O_K=P², P=(5,ω−3).
Here
e=2, f=1, N(P)=5.
This is ramification.
14. Discriminant detects ramified rational primes
For a number field K, a rational prime p ramifies in K exactly when
p | D_K.
For Q(√5), D_K=5, so 5 is the only ramified rational prime.
The examples p=3 and p=11 are unramified, as expected.
15. Split, inert and ramified as polynomial behaviour
In this monogenic quadratic example:
two distinct linear factors → split one irreducible quadratic → inert one repeated linear factor → ramified.
In higher degree, there can be many mixed factorisation patterns, with irreducible factors of several degrees and repeated exponents.
16. Higher-degree example pattern
Suppose a quintic defining polynomial factors modulo p as
f_bar=φ1 φ2²
with
deg φ1=1, deg φ2=2.
Under Dedekind’s index hypothesis,
pO_K=P1 P2²
and
e1f1+e2f2=1·1+2·2=5.
The degree relation checks the full field degree.
17. Completely split primes
A prime p splits completely when
pO_K=P1...Pn
with every e_i=f_i=1.
Then every residue field is F_p.
For Galois extensions, completely split primes have Frobenius element equal to the identity.
18. Totally inert primes
In a degree-n extension, p is inert when pO_K remains prime and the residue degree is n:
e=1, f=n.
The residue field is F_(p^n).
In non-Galois higher-degree fields, the phrase “inert” is usually reserved for this single-prime full-degree case.
19. Totally ramified primes
A prime is totally ramified when
pO_K=P^n
with
e=n, f=1.
Eisenstein polynomials often produce totally ramified local extensions, a connection developed in Guide39.
20. Ramification versus repeated polynomial factors
If O_K=Z[α] locally at p, repeated factors of the minimal polynomial modulo p correspond to ramification.
If p divides the index [O_K:Z[α]], repeated factors can also be artefacts of the bad order.
This is why Guide37’s maximal-order step must come first.
21. Prime ideals as modular constraints
An ideal
P=(p,φ(α))
means two simultaneous congruences:
p≡0, φ(α)≡0.
Reduction modulo P therefore maps algebraic integers into the finite field where α behaves like a chosen root of φ.
This viewpoint is computationally useful when evaluating elements, constructing residue fields and lifting roots.
22. Factorisation of arbitrary ideals
Once rational primes are understood, arbitrary nonzero integral ideals factor uniquely into prime ideals.
If an ideal I has norm
N(I)=∏ p_i^(a_i),
then only prime ideals lying above those rational p_i can divide I.
Ideal factorisation algorithms combine integer norm factorisation with local ideal divisibility tests.
23. Valuations attached to prime ideals
For a nonzero fractional ideal I, write
I=∏_P P^(v_P(I)).
The integer v_P(I) is the P-adic ideal valuation.
For an element α in K*, define
v_P(α)=v_P((α)).
These valuations are the global objects that become local valuations after completion in Guide39.
24. Norms of elements and prime ideals
For α in O_K,
|N_K/Q(α)|=N((α)).
If
(α)=∏P^a_P,
then
|N_K/Q(α)|=∏N(P)^a_P.
This converts ideal valuations into ordinary integer prime exponents weighted by residue degrees.
25. Frobenius in an unramified Galois extension
If K/Q is Galois and P lies above unramified p, the Frobenius automorphism is characterised modulo P by
Frob_P(x) ≡ x^p (mod P).
Its order equals the residue degree f.
Thus prime decomposition connects directly to Galois symmetry.
This is the arithmetic source of Frobenius traces seen later in elliptic curves and Galois representations.
26. Splitting types and Galois groups
In a Galois extension, all primes P_i above an unramified p have the same e and f.
The splitting pattern reflects the conjugacy class of Frobenius.
Chebotarev’s density theorem generalises the idea that modular factorisation patterns are distributed according to Galois-group conjugacy classes.
27. Prime decomposition in algorithms
Prime ideals appear in:
class-group relation collection S-unit computations local completions ideal factor bases in NFS CM and ring-class fields ray class groups valuation-based norm equations.
They are not optional abstractions; they are the working primes of number-field arithmetic.
28. Computing residue degree in software
Computer algebra systems represent prime ideals together with ramification and residue data.
SageMath’s number-field documentation exposes methods for prime ideals above p and residue-class degree. The order documentation also constructs residue fields
O_K/P.
The software output should be checked against
Σe_if_i=n
and norm multiplicativity.
29. Dedekind factorisation receipt
For a prime p and monogenic integral presentation O_K=Z[α], retain:
minimal polynomial f index verification factorisation of f mod p prime ideals (p,φ_i(α)) e_i and f_i norms p^(f_i) degree sum Σe_if_i.
This is enough to independently reconstruct the prime decomposition.
30. Common mistakes
1. Factoring the defining polynomial in a nonmaximal order and declaring the result to be prime-ideal factorisation. 2. Confusing ramification index e with residue degree f. 3. Assuming N(P)=p for every prime above p.
4. Forgetting repeated modular factors. 5. Calling every nonsplit prime inert in higher degree. 6. Forgetting that p dividing D_K is the ramification test, not p dividing an arbitrary polynomial discriminant. 7. Omitting ideal exponents when computing norms. 8. Failing to check Σe_if_i=n.
31. Practice set
1. Define e_i. 2. Define f_i. 3. State Σe_if_i. 4. What is N(P_i)?
5. Factor x²−x−1 mod11. 6. Write the two primes above11. 7. Explain why3 is inert in Q(√5). 8. What is the residue field size above3?
9. Factor x²−x−1 mod5. 10. Give e and f at5. 11. Why is5 the only ramified rational prime? 12. What hypothesis lets Dedekind’s theorem read ideal factorisation from polynomial factorisation?
32. Answers
1. The exponent of P_i in pO_K. 2. [O_K/P_i:F_p]. 3. Σe_if_i=[K:Q]. 4. p^(f_i).
5. (x−4)(x−8). 6. (11,ω−4) and (11,ω−8). 7. The quadratic polynomial has no F_3 root, hence remains irreducible. 8. 9.
9. (x−3)². 10. e=2,f=1. 11. D_K=5. 12. p must not divide [O_K:Z[α]].
Sources and further study
SageMath’s number-field reference documents primes above rational primes and residue-class degrees. Its orders reference constructs residue fields at prime ideals. Standard computational algebraic-number-theory texts develop Dedekind factorisation, ideal norms and index corrections in full detail.
Continue through Batch 10
Return to Guide 37. Continue to Guide 39: Local Fields and Guide 40: S-Units, Norm Equations and Thue Equations.