7.2 Ramification and inertia

Prime decomposition in number fields is a fascinating puzzle. When we extend our number system, primes can split, ramify, or stay inert. This behavior is key to understanding the structure of algebraic integers.

Ramification index and inertia degree are the tools we use to crack this puzzle. They tell us how primes break apart or stay whole, giving us insights into the nature of number field extensions and their arithmetic properties.

Ramification index and inertia degree

Definitions and fundamental concepts

• Ramification index (e) measures the degree of prime ideal splitting or ramification in number field extensions
• Inertia degree (f) represents the residue field extension degree for prime ideals in field extensions
• For prime ideal P in extension L/K above prime ideal p in K, ramification index e(P|p) defined as power of P in pOL factorization
• Inertia degree f(P|p) defined as degree of field extension OL/P over OK/p (OL and OK are rings of integers of L and K)
• Fundamental equality relates ramification index, inertia degree, and field extension degree: $[L:K] = efg$ (g is number of prime ideals in L above p)
• Ramification occurs when $e > 1$, prime inert when $f > 1$ and $e = g = 1$

Applications and importance

• Crucial for understanding prime ideal behavior in algebraic number theory
• Used to analyze factorization of primes in number field extensions
• Helps determine splitting behavior of primes in various types of field extensions (Galois, cyclotomic, quadratic)
• Provides insights into the structure of algebraic integers and ideal class groups
• Utilized in studying L-functions and zeta functions of number fields
• Important in solving Diophantine equations and other number-theoretic problems (Fermat's Last Theorem)

Calculating ramification and inertia

Methods for computing ramification index

• Factor prime ideal p of OK in OL and determine highest power of each prime factor in decomposition
• Use Newton polygons to analyze ramification in extensions of p-adic fields
• Apply Dedekind's theorem on prime ideal factorization in extensions to determine possible e values
• Utilize discriminant calculations and different ideal for ramification information in specific cases
• Employ Kummer's theorem for ramification indices in Kummer extensions (cyclotomic fields)

Techniques for determining inertia degree

• Construct and compare residue fields OL/P and OK/p, determining field extension degree
• Use norm of ideals: $N_{L/K}(P) = p^f$ (p is characteristic of residue field OK/p)
• Apply Galois theory for Galois extensions, utilizing decomposition groups and Frobenius elements
• Analyze splitting field of the reduction of the minimal polynomial modulo p
• Utilize factorization patterns of polynomials over finite fields (irreducible factors)

Advanced calculations and special cases

• For Galois extensions, use relation between decomposition groups, inertia groups, and Frobenius elements
• In towers of field extensions $K \subseteq M \subseteq L$, ramification indices and inertia degrees multiply: $e(P|p) = e(P|Q)e(Q|p)$ and $f(P|p) = f(P|Q)f(Q|p)$ (Q is prime of M between P and p)
• Analyze totally ramified extensions ($e = [L:K]$ and $f = g = 1$) and unramified extensions ($e = 1$ and $f = [L:K]$)
• Study behavior of ramification and inertia in compositum fields and intersections of number fields
• Use local methods (p-adic analysis) for detailed ramification information in specific primes

Ramification, inertia, and prime decomposition

Prime decomposition in number field extensions

• Decomposition of prime ideal p in extension L/K given by $pO_L = (P_1^{e_1} ... P_g^{e_g})$ (Pi are prime ideals in OL, ei are ramification indices)
• Decomposition types: split (g > 1, e = 1), ramified (e > 1), inert (g = 1, e = 1, f > 1)
• Examples of prime decomposition:
  • In $\mathbb{Q}(\sqrt{-5})/\mathbb{Q}$, prime 2 ramifies: $(2) = (2, 1+\sqrt{-5})^2$
  • In $\mathbb{Q}(\sqrt{2})/\mathbb{Q}$, prime 7 splits: $(7) = (7, 3+\sqrt{2})(7, 3-\sqrt{2})$
• Dedekind's theorem provides criteria for prime decomposition based on factorization of minimal polynomial modulo p

Galois theory and decomposition fields

• Decomposition field defined as fixed field of decomposition group (subgroup of Gal(L/K) fixing prime ideal P above p)
• Inertia field defined as fixed field of inertia group (subgroup of decomposition group fixing residue field elements modulo P)
• Frobenius element generates Galois group of residue field extension, order equals inertia degree f
• Decomposition field corresponds to splitting field of reduction of minimal polynomial modulo p
• Example: In $\mathbb{Q}(\zeta_7)/\mathbb{Q}$ ($\zeta_7$ is 7th root of unity), for prime 2:
  • Decomposition field is $\mathbb{Q}(\sqrt{-7})$
  • Inertia field is $\mathbb{Q}$
  • Ramification index e = 1, inertia degree f = 3, g = 2

Ramification and inertia in special extensions

• Cyclotomic fields $\mathbb{Q}(\zeta_n)$: prime p ramifies if and only if p divides n, with known ramification indices
• Quadratic fields $\mathbb{Q}(\sqrt{d})$: prime p ramifies if and only if p divides the discriminant
• Kummer extensions: ramification controlled by congruence conditions on base field elements
• Totally ramified extensions characterized by $e = [L:K]$ and $f = g = 1$ (local fields)
• Unramified extensions have $e = 1$ and $f = [L:K]$ (important in local class field theory)
• Tamely ramified extensions: ramification index not divisible by residue field characteristic
• Wildly ramified extensions: ramification index divisible by residue field characteristic (more complex behavior)

Discriminant of number fields

Definition and properties

• Discriminant of number field K defined as product of squares of differences of conjugates
• Measures extent of ramification in extension K/Q
• For number field K of degree n, discriminant $\Delta_K$ related to different ideal $D_{K/Q}$ by formula: $|\Delta_K| = N_{K/Q}(D_{K/Q})$
• Different ideal $D_{K/Q}$ is product of local differents $D_P$ for all prime ideals P of OK
• For tamely ramified primes: $D_P = P^{e_P-1}$
• For wildly ramified primes: $D_P \geq P^{e_P-1}$ (strict inequality possible)

Calculation methods and formulas

• Ramification contribution to discriminant: $v_p(\Delta_K) = \sum(e_i - 1)f_i$ (sum over primes Pi of OK above p, ei and fi are ramification indices and inertia degrees)
• For Galois extensions, use conductor-discriminant formula relating discriminant to product of Galois group character conductors
• Quadratic fields $\mathbb{Q}(\sqrt{d})$ (d squarefree): $\Delta_K = d$ if $d \equiv 1 \pmod{4}$, $\Delta_K = 4d$ otherwise
• Cyclotomic fields $\mathbb{Q}(\zeta_n)$: $\Delta_K = (-1)^{\phi(n)/2} \frac{n^{\phi(n)}}{\prod_{p|n} p^{\phi(n)/(p-1)}}$ ($\phi(n)$ is Euler's totient function)
• p-adic valuation of discriminant provides information about ramification of p in extension

Applications and interpretations

• Higher discriminant valuations indicate more severe ramification
• Discriminant used to bound class numbers and regulators of number fields
• Important in studying integral bases and defining equations for number fields
• Utilized in algorithms for computing Galois groups and field automorphisms
• Helps in analyzing arithmetic properties of algebraic integers (unique factorization domains)
• Used in formulating and proving various reciprocity laws in algebraic number theory
• Crucial in studying zeta functions and L-functions of number fields (functional equations)