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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.

A field with a surjective discrete valuation, complete for the associated non-Archimedean metric. Its valuation ring is a discrete valuation ring, and its maximal ideal is generated by a uniformizer. Completeness allows successive congruence solutions to converge to actual roots, which underlies the Hensel lemma. Non-Archimedean local fields are examples; completeness alone does not require the residue field to be finite.

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 normalized discrete valuation, a uniformizer $\pi$, and a chosen set $A$ of representatives of the residue field, every existing nonzero element has a unique displayed expansion, with its first digit not in the maximal ideal. Starting from $y_0={\pi }^{-v(x)}x$, choose $a_j$ representing $y_j$ modulo the maximal ideal and put $y_{j+1}=(y_j-a_j)/\pi$. The error after $r$ digits has valuation at least $v(x)+r$, so the sums converge to $x$ even without completeness. Conversely arbitrary digit sequences need not converge in an incomplete field: in $\mathbb{Q}$ with its three-adic valuation, the digits of $\sqrt{10}\in {\mathbb{Q}}_3$ give a counterexample. Hensel lemma gives this square root, but it is not rational.

Let a nontrivially valued local field have compact valuation ring $\mathcal{O}$, and choose any $t$ with $0<|t|<1$. The quotient $\mathcal{O}/t\mathcal{O}$ is finite. A fixed set of representatives containing zero gives every field element a unique convergent digit expansion, with finitely many nonzero negative-index terms. Recursive reduction modulo $t\mathcal{O}$ gives existence; comparing the first differing digit modulo this ideal gives uniqueness. The element $t$ need not be a uniformizer.

For a locally compact field with a nontrivial Non-Archimedean absolute value, scaling a compact neighborhood makes its valuation ring compact. Its open maximal ideal gives a finite discrete residue field, so that ideal is also closed and compact. A nonzero element of maximal absolute value in the ideal generates it. Successive division by this uniformizer makes the value group discrete.

For $n\ge 2$, the normal affine surfacehas $\mathrm{Cl}(X_n)\cong \mathbb{Z}/n\mathbb{Z}$. The class of the prime divisor $D=V(x,z)$ generates: localization at $x$ is a unique factorization domain, so the divisor-class localization sequence leaves only $D$, while $\mathrm{div}(x)=nD$ gives its exact order.

For the three-dimensional affine quadric conethe divisor class group is infinite cyclic. A generator is either ruling plane $V(x,z)$ or $V(x,w)$, and the principal divisor of $x$ gives the relation $[V(x,z)]+[V(x,w)]=0$.

A finite extension $L/K$ of complete discretely valued fields is totally ramified exactly when $L=K(\alpha )$ for a root $\alpha$ of an Eisenstein polynomial over ${\mathcal{O}}_K$. In the reverse direction one may take $\alpha$ to be any uniformizer of $L$.

For a discrete valuation ring with uniformizer $\pi$, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by $\pi$ and whose constant coefficient is not divisible by ${\pi }^2$. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

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.

If a complete discretely valued field $K$ has positive characteristic and perfect residue field $k$, its Teichmuller lift is the unique ring section $k\to {\mathcal{O}}_K$. A choice of uniformizer $t$ then gives

For a characteristic-p local field with residue field ${\mathbb{F}}_q$, the simple roots of $X^q-X$ lift uniquely by Hensel lemma. Their lifts form a copy of ${\mathbb{F}}_q$, since qth powers preserve addition and multiplication. Choose a uniformizer $\pi$. Repeatedly subtracting the lifted residue and dividing by $\pi$ produces a unique convergent expansion ${\sum }_{i\ge 0}{a}_i{\pi }^i$ for each integral element. Allowing finitely many negative powers gives a valued-field isomorphism $K\cong {\mathbb{F}}_q((T))$, $T\mapsto \pi$.

Realize a finite p-group $P$ as the Galois group of a totally ramified extension $E/{\mathbb{F}}_p((t))$, using Artin-Schreier embedding problems over a characteristic-p field. A uniformizer $u$ of $E$ gives $E\cong {\mathbb{F}}_p((u))$: every element has a unique convergent Laurent expansion with coefficients in the residue field. An automorphism has $u\mapsto a_{1}u+a_2{u}^2+\cdots$, and $a_1$ defines a homomorphism $P\to {\mathbb{F}}_p^{\times }$. Its image is trivial because $|P|$ is a $p$-power and $p\nmid |{\mathbb{F}}_p^{\times }|$. Thus the faithful Galois action lies in the Nottingham group, embedding $P$.

For a number field and finite subgroup $\Delta \subset K^{\ast }/K^{\ast n}$, adjoining all the indicated nth roots gives an extension unramified outside the displayed finite set. The valuations modulo $n$ are independent of representatives. At a finite place outside $S$, divide each radicand by a local nth power of a uniformizer to obtain a unit. Its polynomial has separable reduction because $n$ is invertible. A finite unramified extension containing all residue roots contains all lifted roots by Hensel's lemma. This proof does not require ${\mu }_n\subset K$.

Let $m_1,m_2$ be coprime positive square-free integers, with $m_1\not\equiv \pm m_2(\mathrm{mod}9)$. The displayed integral basis has discriminant $-27m_1^2{m}_2^2$. Both nonconstant generators are integral, and their products are integral linear combinations of the three basis elements. At a prime dividing $m_1$, the cubic for $\alpha$ is Eisenstein; at a prime dividing $m_2$, the cubic for ${\alpha }^2/m_2$ is Eisenstein. If $3\nmid m_1{m}_2$, then $m_1{m}_2^2\not\equiv \pm 1(\mathrm{mod}9)$, and a shift by $1$ or $-1$ gives an Eisenstein polynomial at three. These local uniformizer generators show that the order is maximal at all primes dividing its discriminant; elsewhere the discriminant-index formula for an integral lattice already gives maximality.

For a p-modular system $(K,\mathcal{O},k)$, an integral form of a finite-dimensional $KG$-module $V$ is a $G$-stable finite free $\mathcal{O}$-submodule $W\subset V$ with $K{\otimes }_{\mathcal{O}}W\cong V$. Starting with a basis lattice $L$, the sum ${\sum }_{g\in G}gL$ supplies such a form: it is finitely generated and torsion-free, hence free over the discrete valuation ring $\mathcal{O}$. Reduction gives the $kG$-module $W/\pi W$, where $\pi$ is a uniformizer.

The local Artin map is the reciprocity homomorphism of Local Artin reciprocity. Its normalization is fixed by choosing whether a uniformizer maps to arithmetic or geometric Frobenius.

Local class field theory describes the abelian extensions of a local field $K$ through its multiplicative group $K^{\times }$.

For a finite Galois extension $L/K$ of non-Archimedean local fields, its relative Brauer group is the kernel of restriction $\mathrm{Br}(K)\to \mathrm{Br}(L)$ and identifies with $H^2(\mathrm{Gal}(L/K),L^{\times })$. Restriction multiplies the local Brauer invariant by $[L:K]$. This kernel is therefore cyclic of order $[L:K]$, and the indicated class is its distinguished generator. In an unramified extension with arithmetic Frobenius generator it is represented by the cyclic algebra with parameter a uniformizer.