= Solution
Let $L/K$ be a finite <Galois extension> of non-Archimedean <local fields>, with <Galois group> $G$, normalized <valuation> $v_L$, integer ring $\mathcal O_L$, and residue characteristic $p$. The <lower ramification numbering> is
$$
G_{-1}=G,\qquad
G_i=\{\sigma\in G:v_L(\sigma a-a)\ge i+1
\text{ for every }a\in\mathcal O_L\},\quad i\ge0.
$$
These <higher ramification groups> measure successively finer agreement with the identity. The <group> $G_0$ is the <inertia group>, the kernel of the action on the <residue field>. Each $G_i$ is normal in $G$: conjugation permutes integral elements and preserves their <valuations>. The <groups> eventually become trivial, since each nonidentity automorphism moves some integral element and its displacement has finite <valuation>.
Let $K_0=L^{G_0}$, the <maximal unramified subextension of a local field extension>. Then $G/G_0$ is the <Galois group> of the residue extension, cyclic and generated by a <Frobenius automorphism>. For a <uniformizer> $\pi$ of $L$, the <uniformizer criterion for lower ramification groups> gives
$$
G_i=\{\sigma\in G_0:v_L(\sigma\pi-\pi)\ge i+1\}.
$$
To see why one <uniformizer> suffices, the extension $L/K_0$ is totally ramified and its integer ring is $\mathcal O_{K_0}[\pi]$. Elements of $G_0$ fix its <coefficients>. Differences of powers of $\pi$ are divisible by $\sigma\pi-\pi$, so the <valuation> bound for $\pi$ implies it for every integral element. The converse follows by testing $\pi$ itself.
The <tame inertia quotient> is seen explicitly from
$$
G_0\longrightarrow k_L^\times,\qquad
\sigma\longmapsto\overline{\sigma\pi/\pi}.
$$
It is a homomorphism because inertia acts trivially on the <residue field>, and its kernel is $G_1$. Thus $G_0/G_1$ is cyclic of order not divisible by $p$. For $i\ge1$, the homomorphism
$$
G_i\longrightarrow(k_L,+),\qquad
\sigma\longmapsto
\overline{\frac{\sigma\pi-\pi}{\pi^{i+1}}}
$$
has kernel $G_{i+1}$. Indeed composing two automorphisms adds the leading displacements, since $\sigma\pi/\pi\equiv1$ and inertia fixes residue <coefficients>. Hence each $G_i/G_{i+1}$ is an <elementary abelian p-group>. Eventual triviality makes $G_1$ a <p-group>; the prime-to-p quotient makes it the unique <Sylow subgroup> of $G_0$. This is the <wild inertia group>. <Tame ramification> means $G_1=1$.
The filtration also measures the <different exponent>. For the <totally ramified extension> over $K_0$, the <minimal polynomial of an algebraic element> $g$ of $\pi$ has <polynomial roots> $\sigma\pi$, $\sigma\in G_0$. The derivative formula gives
$$
v_L(g'(\pi))=\sum_{\sigma\in G_0\setminus\{1\}}
v_L(\pi-\sigma\pi).
$$
For a fixed $\sigma$, its displacement <valuation> is the number of indices $i\ge0$ for which it belongs to $G_i$. Interchanging the finite sums therefore gives <different exponent from ramification groups>:
$$
\boxed{\delta(L/K)=\sum_{i\ge0}(|G_i|-1).}
$$
The unramified part contributes zero. More generally <different in a tower> follows by taking products of <bases of a module> dual for the <trace pairing>, and gives
$$
\delta(L/K)=\delta(L/E)+e(L/E)\delta(E/K).
$$
Thus a tame extension has <different exponent> $e-1$, while nontrivial wild <groups> contribute additional positive terms.
The lower numbering restricts to <subgroups>: if $H=\operatorname{Gal}(L/E)$, its <groups> are $H_i=H\cap G_i$, directly from the definition using the same top-field <valuation>. Quotients need a different indexing. For real $s\ge0$ put $G_s=G_{\lceil s\rceil}$ and define the <Herbrand function>
$$
\varphi(s)=\int_0^s\frac{dt}{[G_0:G_t]},\qquad
\psi=\varphi^{-1},\qquad G^u=G_{\psi(u)}.
$$
This <upper ramification numbering> stretches or compresses intervals according to the inertia indices. The quotient theorem says that for a <normal subgroup> $H$, <upper ramification groups commute with quotients>:
$$
(G/H)^u=G^uH/H.
$$
In particular the same upper index has a meaning compatible with passing to a smaller <Galois extension>, which lower indices generally lack.
For a tame <totally ramified extension> the picture is simple: $G_0=G$, all $G_i$ with $i\ge1$ are trivial, and the <different exponent> is $|G|-1$. The extension $\mathbb Q_p(\zeta_p)$ for odd $p$ illustrates this, giving exponent $p-2$.
A mixed-characteristic wild family is the <lower ramification filtration of a prime-power cyclotomic extension>. Take $L=\mathbb Q_p(\zeta_{p^r})$ with odd $p$ and $\pi=\zeta_{p^r}-1$. If $\sigma_a(\zeta)=\zeta^a$ and $a\ne1$ modulo $p^r$, then
$$
v_L(\sigma_a\pi-\pi)
=v_L(\zeta^{a-1}-1)=p^{v_p(a-1)}.
$$
The last equality follows because $\zeta^{a-1}$ has order $p^{r-v_p(a-1)}$ and its <uniformizer> <valuation> scales by the <ramification index> in that cyclotomic tower. The lower breaks are $0,p-1,p^2-1,\ldots,p^{r-1}-1$. Integrating the successive indices $(p-1),p(p-1),\ldots$ gives upper breaks $0,1,\ldots,r-1$.
In equal characteristic there is an equally explicit wild example. Start with $K=\mathbb F_q((t))$ and an <Artin–Schreier extension>
$$
y^p-y=t^{-m},\qquad m>0,\quad p\nmid m.
$$
There is no <polynomial root> in $K$: a negative-valuation element has $v(x^p-x)=p\,v(x)$, while the right side has <valuation> $-m$. Adjoining a <polynomial root> splits the <polynomial>, whose <polynomial roots> are $y+c$ for $c\in\mathbb F_p$, so the extension is cyclic of degree $p$. The <valuation> equation forces total ramification, $v_L(t)=p$ and $v_L(y)=-m$. Choose $1\le b\le p-1$ with $-bm\equiv1\pmod p$ and set $a=(1+bm)/p$. Then $\pi=t^a y^b$ is a <uniformizer>. For a nonidentity automorphism $y\mapsto y+c$,
$$
\frac{\sigma\pi}{\pi}=(1+c/y)^b,\qquad
v_L(\sigma\pi-\pi)=1+m;
$$
the first <binomial coefficient> $b$ is nonzero modulo $p$, and all subsequent terms have strictly higher <valuation>. Therefore
$$
G_0=\cdots=G_m=C_p,\qquad G_{m+1}=1,\qquad
\delta=(m+1)(p-1).
$$
This is the <ramification break of an Artin–Schreier pole>. It shows concretely that wild breaks, unlike tame inertia, can occur arbitrarily far out in the filtration.
\b[The lower <groups> measure integral displacements, the upper <groups> make quotient comparison possible, and the sizes of the <groups> determine the different.]
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