Ramification group ($G_i$)

For a finite Galois extension of local fields, the lower ramification groups measure how closely automorphisms fix the valuation ring. The inertia group is $G_0$, and $G_1$ is the wild inertia group.

For $\sigma \in G_i$, the residue of $(\sigma {\pi }_L-{\pi }_L)/{\pi }_L^{i+1}$ gives a homomorphism to the additive residue field with kernel $G_{i+1}$. On composing two automorphisms, their leading displacement coefficients add because each fixes residues and changes the uniformizer only in degree at least two. These successive quotients are elementary abelian groups in residue characteristic $p$. The filtration terminates, so the wild inertia group is a finite $p$-group.

For an odd prime $p$, this Eisenstein polynomial generates a totally ramified cubic root field. Its discriminant is $-p^2(4p+27)$. For $p>3$ the unit part is a square exactly when $p\equiv 1(\mathrm{mod}3)$. The splitting field then has cyclic Galois group of order three; otherwise it has symmetric group $S_3$, with an unramified extension of degree two and inertia group $C_3$. Both cases are tamely ramified, so all positive ramification groups are trivial.

At $p=3$, adjoining $\beta =\sqrt{-39}$ gives a ramified quadratic subfield, and the splitting field has total ramification index six. A root $\alpha$ has normalized valuation two, while $\beta$ has valuation three, so $\pi =\beta /\alpha$ is a uniformizer. The derivative $3{\alpha }^2+3$ has valuation six, and all distinct root differences have valuation three. A three-cycle fixes $\beta$, giving $v_L(\sigma \pi -\pi )=2$. A transposition fixing $\alpha$ negates $\beta$, giving valuation one; conjugation handles the other transpositions. Hence $G_0=S_3$, $G_1=C_3$, and $G_i=1$ for $i\ge 2$. The different exponent is seven.

For $L={\mathbb{Q}}_p({\zeta }_{p^n})$, the uniformiser ${\zeta }_{p^n}-1$ and the uniformizer criterion for lower ramification groups give the displayed valuation for $a\ne 1$. Thus $G_0$ is the full unit group modulo $p^n$, and $G_i$ consists of units congruent to one modulo $p^j$ when $p^{j-1}\le i\le p^j-1$, $1\le j<n$. All groups are trivial for $i\ge p^{n-1}$. The formula includes $p=2$, where the first congruence subgroup is already the full group.

For all real $s\ge 0$, this definition uses the normalized integral valuation on $L$. Equivalently $G_s=G_{\lceil s\rceil }$. The group $G_0$ is the inertia group and $G_1$ is the wild inertia group.

The Herbrand function is a continuous strictly increasing piecewise linear function on the nonnegative reals. Its inverse is denoted ${\psi }_{L/K}$. It reparametrizes the lower ramification numbering to the upper ramification numbering.

The upper numbering is obtained by applying the inverse Herbrand function to the index in the lower ramification numbering. It is compatible with quotient Galois groups, whereas lower numbering is compatible with subgroups.

For a normal subgroup $H$ of a finite local Galois group, the upper ramification numbering on its quotient is obtained by projecting the upper groups. The Herbrand function is the change of variable that ensures this compatibility. In contrast, the lower ramification numbering restricts directly to subgroups.

If $L/K$ is a finite totally ramified Galois extension and ${\pi }_L$ is a uniformizer, thenfor every $i\ge 0$.

For $L={\mathbb{Q}}_3({\zeta }_3,\alpha )$ with ${\alpha }^3=2$, the shifted cubic for $\alpha +1$ and quadratic for ${\zeta }_3-1$ are Eisenstein. Their compositum is totally ramified of degree six. The element $\pi =({\zeta }_3-1)/(\alpha +1)$ has valuation one. Order-three automorphisms move it with valuation two and transpositions with valuation one. The lower breaks are zero and one, the upper breaks zero and one-half, and the different exponent from ramification groups is seven.

For ${\mathbb{Q}}_2({\zeta }_8)/{\mathbb{Q}}_2$, the shifted polynomial $(1+T{)}^4+1$ is Eisenstein and ${\zeta }_8-1$ is a uniformizer. The three nonidentity automorphisms have displacement valuations two, four and two. Thus the lower breaks are one and three, the upper breaks one and two, and the different exponent from ramification groups is eight.

Let $K$ be an equal-characteristic-p local field and let $c\in K$ have $v_K(c)=-m<0$, with $p\nmid m$. If ${\alpha }^p-\alpha =c$, the Artin–Schreier extension $L=K(\alpha )$ has degree $p$ and is totally ramified: a root in $K$ would force the pole order to be divisible by $p$, and $p{v}_L(\alpha )=-me(L/K)$ forces $e=p$. Choose $1\le s<p$ and an integer $r$ with $rp-sm=1$. For a base uniformizer $t$, the element ${\pi }_L=t^r{\alpha }^s$ has valuation one. Under ${\sigma }_a(\alpha )=\alpha +a$, $a\in {\mathbb{F}}_p^{\times }$, the leading term in ${\sigma }_a({\pi }_L)-{\pi }_L$ has valuation $rp-(s-1)m=m+1$. The uniformizer criterion for lower ramification groups therefore gives the single lower break $m$. The Herbrand function is the identity up to that break, so the upper break is also $m$.

For the splitting field $M/{\mathbb{Q}}_3$ of $T^3-3T+3$, the cubic is Eisenstein and its discriminant is $-135$. The quadratic subfield is ${\mathbb{Q}}_3(\sqrt{3})$, and the degree-six extension is totally ramified. If $\alpha$ is a root and $s=\sqrt{3}$, then $\pi ={\alpha }^2/s$ is a uniformizer. Transpositions have $v_M(\sigma \pi -\pi )=1$, while three-cycles have value $2$, giving the displayed ramification groups. Its different exponent from ramification groups is $7$.

For odd $p$, the splitting field $K={\mathbb{Q}}_p({\zeta }_p,\sqrt[p]{p})$ is totally ramified of degree $p(p-1)$. Withthe lower ramification groups are

To verify the filtration, write ${\alpha }^p=p$ and $\lambda ={\zeta }_p-1$. The Eisenstein polynomials for $\alpha$ and $\lambda$ give coprime degrees $p$ and $p-1$, so their compositum has degree $p(p-1)$. Its element $\varpi =\lambda /\alpha$ has $p$-adic valuation $1/[p(p-1)]$, forcing total ramification. For an automorphism ${\sigma }_{a,b}$ with $\alpha \mapsto {\zeta }_p^a\alpha$ and ${\zeta }_p\mapsto {\zeta }_p^b$,Its residue is $b$, so the normalized displacement valuation is one if $b\ne 1$. If $b=1$ and $a\ne 0$, that valuation is $1+v_K({\zeta }_p^{-a}-1)=p+1$. The uniformizer criterion for lower ramification groups now gives every group displayed above.

If $a^3=3$ and ${\zeta }_3$ is a primitive cube root of unity, then ${\pi }^3=1+2{\zeta }_3$ and ${\pi }^6=-3$. The Eisenstein polynomial $X^6+3$ therefore presents the entire degree-six splitting field, and $\pi$ is a uniformizer. Its normalised valuation gives $v_L(3)=6$, $v_L(a)=2$ and $v_L({\zeta }_3-1)=3$. The order-three automorphisms displace $\pi$ with valuation four, whereas transpositions displace it with valuation one.

For a finite totally ramified Galois extension, the mapinduces an injection $G_0/G_1\hookrightarrow k_L^{\times }$. In particular, the tame inertia quotient is cyclic and has order prime to the residue characteristic.

For a Finite Galois extension of $K(T)$ with algebraically closed characteristic-zero constants, the local decomposition and inertia groups coincide and are cyclic. A fixed compatible choice of primitive roots of unity selects a generator through its uniformizer character of inertia. Changing the prime above a base point conjugates this generator, and changing the uniformizer does not change its character. Without the roots-of-unity convention, only the inertia-subgroup conjugacy class is intrinsic. Over $\mathbb{C}$, positive local orientation gives the standard convention.

In characteristic zero with algebraically closed constants, a local unit has a square root. Thus $K(T)(\sqrt{a})$ near a point is equivalent to adjoining the square root of the local parameter to the power $v_p(a)$. Even order is unramified; odd order gives ramification index two. For a polynomial $a$, its order at infinity is minus its degree, so infinity branches exactly for odd degree.

For a finite Galois extension of non-Archimedean local fields, this is a homomorphism from the inertia group to $k_L^{\times }$ with kernel the wild inertia group. Inertia acts trivially on residue units, which proves multiplicativity and independence of the chosen uniformizer. Passing to the maximal unramified subextension reduces the kernel calculation to a totally ramified extension. Hence it embeds the tame inertia quotient into the cyclic multiplicative group of the residue field.

For a finite separable extension of local fields, the different is the inverse fractional ideal of the codifferent. If ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$ and $f$ is the minimal polynomial of $\alpha$, then ${\mathfrak{D}}_{L/K}=(f^{\prime }(\alpha ))$.

The different exponent at a prime ideal $\mathfrak{P}$ is its exponent in the prime ideal factorization of the different ideal. It equals the local different exponent of the corresponding extension of completions. The different exponent and tame ramification theorem detects unramified, tame, and wild behavior without assuming a Galois extension.

For $K_r={\mathbb{Q}}_p({\zeta }_{p^r})$, the uniformizer ${\zeta }_{p^r}-1$ has different exponent $p^{r-1}(r(p-1)-1)$. Differentiate the cyclotomic polynomial at ${\zeta }_{p^r}$: the numerator contributes $r{p}^{r-1}(p-1)$ to the valuation, while the denominator ${\zeta }_p-1$ contributes $p^{r-1}$.

For a finite extension of number fields, the different exponent satisfies $d_{\mathfrak{P}}\ge e_{\mathfrak{P}}-1$, with equality exactly when the extension at $\mathfrak{P}$ is tamely ramified. The finite residue fields are perfect, so tameness is equivalent to the residue characteristic not dividing the ramification index. Thus unramified primes have exponent zero and wildly ramified primes have exponent at least $e_{\mathfrak{P}}$. No Galois extension hypothesis is needed.

For a finite separable extension of number fields or local fields,It is also called the codifferent. If ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$ with minimal polynomial $g$, it equals $g^{\prime }(\alpha {)}^{-1}{\mathcal{O}}_L$.

For finite separable extensions $L/E/K$,For normalized valuations this says

For a finite Galois extension of local fields,

The inertia group of a finite Galois extension of local fields is the kernel of the action of its Galois group on the residue field. In lower numbering it is the zeroth ramification group $G_0$.

The wild inertia group is the first ramification group $G_1$. It is the unique Sylow subgroup of the inertia group for the residue characteristic.

A finite extension of local fields is tamely ramified when its residue-field extension is separable and its ramification index is coprime to the residue characteristic. For a finite Galois extension this is equivalent to trivial wild inertia group $G_1$.

If $L/K$ is a finite tamely ramified Galois extension of local fields with degree $n$ and ramification index $e$, then its different exponent is $e-1$ and the discriminant exponent is

If $L/K$ is a finite Galois extension of local fields, ${\mathcal{O}}_L={\mathcal{O}}_K[\alpha ]$, and $f$ is the minimal polynomial of $\alpha$, then