Arithmetic

Arithmetic studies elementary operations on numbers.

The geometric mean of finitely many nonnegative real numbers is the nonnegative $n$th root of their product. For two positive numbers it is $\sqrt{ab}$. The arithmetic-geometric mean inequality compares it with their arithmetic mean. Unlike an arithmetic average, the geometric mean respects multiplicative scaling and is natural for growth factors.

Starting from $0<a_1\le b_1$, replace the lower endpoint by the geometric mean and the upper endpoint by the arithmetic mean. The arithmetic-geometric mean inequality gives $a_n\le a_{n+1}\le b_{n+1}\le b_n$. The bounded monotone sequence theorem gives limits, and the arithmetic recurrence forces them to coincide. The limiting value is the arithmetic-geometric mean of the initial pair.

The natural numbers are the nonnegative integers $0,1,2,\dots$.

Every nonempty subset of the natural numbers has a least element in their usual order. Applying this principle to positive integer linear combinations proves Bézout's identity. This principle for the usual natural-number order is distinct from the general well-ordering theorem asserting that every set can be equipped with some well-order.

A real number is an element of the complete ordered field $\mathbb{R}$.

The Archimedean property says that the natural numbers are unbounded in the real numbers. In particular, for every positive real number $a$ there is a positive integer $n$ with $1/n<a$. Otherwise a purported upper bound for all natural numbers contradicts the least-upper-bound property by translating that bound by one.

A binary number is a number represented in base two.

Every real number in $[0,1]$ is a sum ${\sum }_{j\ge 1}{\epsilon }_j{2}^{-j}$ with binary digits ${\epsilon }_j\in \{0,1\}$.

A dyadic rational has the form $m/2^k$. Its terminating binary expansion has an alternative expansion ending in infinitely many $1$ digits.

The extended real numbers adjoin $-\infty$ and $+\infty$ to $\mathbb{R}$. They preserve the total order but leave some arithmetic expressions, such as $+\infty -\infty$, undefined.

Addition combines two numbers or elements of an additive algebraic structure into their sum.

The additive inverse $-x$ satisfies $x+(-x)=0$.

Summation is repeated addition, conventionally written with the symbol $\sum$.

A sum combines a finite or infinite sequence of terms by repeated addition. The symbol $\sum$ denotes summation over an index.

For positive weights $w_i$, the weighted mean isIt lies between the smallest and largest $x_i$; the same result holds for integrals with a positive weight function.

The arithmetic mean of $x_1,\dots ,x_n$ isIt is the weighted mean with equal weights.

The harmonic mean of positive numbers $x_1,\dots ,x_n$ is $H=n/({\sum }_{i}1/x_i)$. The Jensen inequality for $1/x$ gives $H\le \overline{x}$, with equality exactly when all entries coincide.

The prefix sums $P_m={\sum }_{i=1}^m{a}_i$ can all be computed by the recurrence $P_m=P_{m-1}+a_m$ in linear time. Any contiguous sum is then $P_b-P_{a-1}$, so it takes constant time after preprocessing.

Subtraction adds the additive inverse: $a-b=a+(-b)$.

Multiplication combines two factors into their product.

For a nonnegative integer exponent, exponentiation iterates multiplication: $a^0=1$ and $a^{b+1}=a{a}^b$. This convention defines $0^0=1$ in primitive recursive arithmetic. Extensions to other exponents may use different domains and conventions.

The hyperoperation hierarchy successively iterates basic arithmetic operations. In the usual indexing, successor is followed by addition, multiplication, exponentiation and their higher iterates. A sequence beginning with addition shifts this index by one. Each fixed rank is a primitive recursive function; this does not imply that the evaluator taking the rank as an additional variable is primitive recursive.

A product over the empty set has value one, the identity for multiplication. This makes splitting a product into successive pieces valid even when one piece contains no factors. In particular, the factorial convention is $0!=1$.

A multiplicative inverse of $x$ is an element $x^{-1}$ satisfying $x{x}^{-1}=x^{-1}x=1$.

An arithmetic progression is a sequence $a,a+r,\dots ,a+(m-1)r$ with constant successive difference $r$. It is nonconstant when $r\ne 0$.

If all residues of a modular arithmetic progression have integer representatives in an interval of diameter less than half the modulus, the representatives form an integer arithmetic progression. Consecutive differences lie in an interval shorter than the modulus, so the common modular difference has a unique possible integer representative.

The common difference of an arithmetic progression is the fixed difference between consecutive terms.

Non-Archimedean analysis studies fields whose distance is induced by an ultrametric absolute value, especially local fields and valued function fields.

The continuous functions from the p-adic integers to the P-adic numbers form a Banach space over ${\mathbb{Q}}_p$ with supremum norm $\Vert f{\Vert }_{\infty }={\sup }_{x\in {\mathbb{Z}}_p}|f(x){|}_p$. Compactness of the domain bounds the norm, and uniform convergence in the complete codomain proves completeness. Translation and the forward difference operator $\Delta f(x)=f(x+1)-f(x)$ are bounded operators, with $\Vert \Delta f{\Vert }_{\infty }\le \Vert f{\Vert }_{\infty }$.

Every function in continuous functions on the p-adic integers has a unique uniformly convergent expansion in the binomial polynomials, with $a_n\to 0$ and $a_n=({\Delta }^{n}f)(0)$. Conversely, any coefficient sequence tending to zero defines such a continuous function. To prove the expansion once $a_n\to 0$ is known, use $|(\genfrac{}{}{0ex}{}{x}{n}){|}_p\le 1$ for uniform convergence. Finite binomial inversion gives agreement with $f$ at each nonnegative integer, and density gives agreement everywhere. The coefficients recover successively from these integer values, proving uniqueness. Moreover $\Vert f{\Vert }_{\infty }={\sup }_n|a_n{|}_p$, because finite differences bound each coefficient by the norm and the expansion gives the reverse inequality.

The Mahler coefficients tend to zero, so the displayed series defines a continuous function and has value zero at zero. The Pascal's identity and uniform convergence give $\Delta Jf=f$. Thus the forward difference operator is surjective on continuous functions on the p-adic integers. Its kernel consists of constants: period one implies agreement on the dense nonnegative integers, and continuity then implies constancy. This selects the unique discrete antiderivative vanishing at zero.

There is also a direct construction on locally constant functions. If $h$ has period $M=p^r$, put $a_j=h(j)$, $S={\sum }_{i=0}^{M-1}{a}_i$, and $P_j={\sum }_{i<j}{a}_i$. For $x=j+My$ with $0\le j<M$ and $y\in {\mathbb{Z}}_p$, setThis is a continuous function on each residue class. Increasing $j$ gives $Jh(x+1)-Jh(x)=a_j$; at $j=M-1$, the next point has residue zero and quotient $y+1$, giving the same identity. The ultrametric inequality gives $\Vert Jh{\Vert }_{\infty }\le \Vert h{\Vert }_{\infty }$, and $Jh(0)=0$. The construction is independent of the chosen period, since two normalized discrete antiderivatives agree on the nonnegative integers and then on the p-adic integers by continuity. It is linear on the locally constant functions, which form a dense subset in the supremum norm. Completeness therefore extends it to every continuous function, retaining $\Delta J=I$ and the norm bound. This proves surjectivity without first using the Mahler theorem.

Let $T:\mathcal{C}({\mathbb{Z}}_p,{\mathbb{Q}}_p)\to {\mathbb{Q}}_p$ be linear and invariant under translation by one. For any $f$, choose its continuous discrete antiderivative $Jf$. Then $T(f)=T(\Delta Jf)=T(Jf(\cdot +1))-T(Jf)=0$. No continuity or boundedness of $T$ is assumed. In particular this applies to a form invariant under every translation by a p-adic integer.

For a continuous function $f:{\mathbb{Z}}_p\to {\mathbb{Q}}_p$, its nth Mahler coefficient isThis is the explicit iterate of the forward difference operator. These coefficients are exactly those of its Mahler expansion.

Approximate a continuous function on ${\mathbb{Z}}_p$ uniformly by a function constant on residue classes modulo $p^r$. On this finite space, translation $S$ satisfies $S^{p^r}=I$, so ${\Delta }^{p^r}=(S-I{)}^{p^r}$ has every matrix coefficient divisible by $p$. Hence $\Vert {\Delta }^{j{p}^r}\Vert \le p^{-j}$, and the Mahler coefficients of a locally constant function tend to zero. Since $|a_n(f)-a_n(g){|}_p\le \Vert f-g{\Vert }_{\infty }$, uniform approximation proves the assertion for every continuous function. This completes the coefficient-decay part of the Mahler theorem.

A non-Archimedean ordered field contains elements larger than every positive integer, equivalently positive infinitesimals smaller than every positive rational number.

An absolute value on a field is a nonnegative multiplicative function that vanishes only at zero and satisfies the triangle inequality.

The metric completion of a field for an absolute value on a field is formed from Cauchy sequences modulo sequences tending to zero. Addition and multiplication extend continuously, and every nonzero completed element is invertible, so the result remains a field. The absolute value on a field extends uniquely. A place of a number field determines the corresponding completion $K_v$.

The trivial absolute value on a field is zero at zero and one elsewhere. Its metric topology is discrete. Its valuation ring and residue field are both the field itself. This is why rational absolute-value classification and nontrivial discrete-value compactness criteria need a nontriviality hypothesis.

A non-Archimedean absolute value satisfies the ultrametric inequality $|x+y|\le \max (|x|,|y|)$.

An absolute value on a field is non-Archimedean exactly when its values on the image of $\mathbb{Z}$ are bounded. The reverse implication follows by applying the ordinary triangle inequality to the binomial expansion of $(x+y{)}^n$ and taking $n$th roots as $n\to \infty$.

The ultrametric inequality is $d(x,z)\le \max (d(x,y),d(y,z))$. For unequal $|x|$ and $|y|$, it implies $|x+y|=\max (|x|,|y|)$.

An ultrametric space is a metric space satisfying the ultrametric inequality. Its open balls are clopen, and any two balls are either disjoint or one contains the other.

A topological space is totally disconnected when its only connected subsets are singletons.

Two nontrivial absolute values on a field are equivalent when they induce the same topology. Equivalently, one is a positive real power of the other.

For a field extension $L/K$, an extension of an absolute value $|\cdot |$ on $K$ is an absolute value on $L$ whose restriction to $K$ is $|\cdot |$.

If $K$ is complete and $L/K$ is finite, an absolute value on $K$ extends uniquely to $L$. With the normalization that it restricts exactly to the given absolute value,Every $K$-automorphism of $L$ preserves this extended absolute value.

On a finite-dimensional vector space over ${\mathbb{Q}}_p$, the triangle inequality gives an upper bound by a coordinate maximum norm. The coordinate unit sphere is compact. The other norm is continuous and nonzero there, so has a positive minimum. Scaling by powers of $p$ gives the lower bound. The resulting equivalence of norms in finite dimensions proves completeness of finite extensions, and applying the comparison to powers of a field element proves equality of two extending multiplicative absolute values on a field.

Suppose a Non-Archimedean absolute value extends uniquely to every finite extension. If $f\in {\mathcal{O}}_K[X]$ is monic and irreducible, every root has absolute value at most one: otherwise its leading power dominates all lower terms, contradicting $f(\alpha )=0$ by the ultrametric inequality. In a splitting field, let $\phi$ be the minimal polynomial of an algebraic element of the residue of one root. Lift its coefficients to ${\mathcal{O}}_K$. The lift has absolute value less than one at that root and hence, by equal absolute values of algebraic conjugates, at every root. Consequently every root residue is a zero of $\phi$. Since reduction preserves the product of the linear factors with multiplicities, every irreducible factor of $\overline{f}$ is $\phi$, proving the displayed identity for some $m\ge 1$.

Suppose a Non-Archimedean absolute value on $K$ has a unique extension to every finite field extension. Any K-isomorphism $K({\alpha }_i)\to K({\alpha }_j)$ between conjugate roots pulls the absolute value back to an extension on the source field, which must be the given one. Thus $|h({\alpha }_i)|=|h({\alpha }_j)|$ for every $h\in K[X]$. This argument includes inseparable irreducible polynomials: no transitive Galois action or completeness assumption is needed.

Let $K$ be complete for a discrete valuation and let $L/K$ be finite. The valuation ring ${\mathcal{O}}_L$ is exactly the integral closure of ${\mathcal{O}}_K$ in $L$. One direction follows from a monic integral equation; for the other, the coefficients of the minimal polynomial are elementary symmetric polynomials in conjugates of absolute value at most one.

If $L/K$ is a finite normal extension, the Galois group acts transitively on the absolute values of $L$ extending a fixed absolute value of $K$. For number fields, this is equivalent to transitivity on the prime ideals above a fixed prime.

Give $\mathbb{Q}$ its $5$-adic valuation, take $L=\mathbb{Q}(i)$, and choose the extension associated with the prime $(2+i)$. Then $(2+i)/(2-i)$ lies in the chosen valuation ring but has negative valuation at the conjugate prime. It is therefore not in the integral closure of ${\mathbb{Z}}_{(5)}$ in $L$.

If $L=K(\alpha )$ is a finite separable extension and $v$ is an absolute value on $K$, the extensions of $v$ to $L$ correspond to the irreducible factors of the minimal polynomial of $\alpha$ over the completion of $K$ at $v$.

This polynomial is Eisenstein at $2$. Over ${\mathbb{Q}}_3$, its Newton polygon has a single slope $-2/5$, so it is irreducible and a root generates a totally ramified extension of degree five. Over ${\mathbb{Q}}_7$, its factor degrees are $1,1,3$: the simple residue roots $1,-1$ lift, and the remaining roots have valuation $1/3$. Thus the degree-five number field generated by a root has one prime above $3$ and three above $7$.

For a finite extension of number fields $L/K$ and a nonzero prime ideal $\mathfrak{p}$ of $K$, the primes of $L$ above $\mathfrak{p}$ correspond to the orbits ofAn embedding determines a prime by pulling back the local absolute value on a field. Embeddings with the same prime extend to embeddings of the same completion and are conjugate. If $L=K(\alpha )$, these orbits are the irreducible factors of its minimal polynomial over $K_{\mathfrak{p}}$.

For a number field $K$ and a nonzero prime ideal $\mathfrak{p}\subset {\mathcal{O}}_K$, the completion $K_{\mathfrak{p}}$ is the completion for the $\mathfrak{p}$-adic absolute value. It is a finite extension of ${\mathbb{Q}}_p$, where $p\mathbb{Z}=\mathfrak{p}\cap \mathbb{Z}$.

For a number field $K$ and a rational prime $p$, the natural diagonal map gives an isomorphism of ${\mathbb{Q}}_p$-algebras

If $L/K$ is a finite Galois extension of number fields, its Galois group acts transitively on the prime ideals of ${\mathcal{O}}_L$ lying above a fixed prime ideal of ${\mathcal{O}}_K$.

For a prime $\mathcal{P}$ of a finite Galois extension $L/K$ above $\mathfrak{p}$, the decomposition group is the stabilizer

Let $M/K$ be a finite Galois extension, $L=M^H$, and $D$ the decomposition group of a prime $\mathfrak{P}$ above $\mathfrak{p}$. The bijectionfollows from transitivity on primes for both $M/K$ and $M/L$.

If $M/K$ is the Galois closure of a finite extension $L/K$, a prime splits completely in $L$ exactly when it splits completely in $M$. Complete splitting means every right action of $D$ on $H\backslash G$ fixes each coset, so $D$ lies in the subgroup core of $H$. That core is trivial because the conjugates of $L$ generate $M$.

Restriction to completions gives a canonical isomorphismConsequently the number of primes above $\mathfrak{p}$ is the index of the decomposition group.

For a prime $\mathfrak{q}$ of a finite separable extension $L/K$ above $\mathfrak{p}$, completion preserves the ramification index and residue-field extension:

The polynomialsplits into three linear factors over ${\mathbb{Q}}_2$. Over ${\mathbb{Q}}_5$ it is an Eisenstein polynomial, so adjoining a root gives a totally ramified cubic extension. Its splitting field over ${\mathbb{Q}}_5$ has decomposition group $S_3$ and inertia group $A_3\cong C_3$.

For an ordered abelian group $\Gamma$, the Hahn series field $k((t^{\Gamma }))$ consists of formal sums with well-ordered support in $\Gamma$. The least exponent defines a valuation with value group $\Gamma$.

A $p$-adic number is an element of the completion ${\mathbb{Q}}_p$ of the rational numbers with respect to the $p$-adic absolute value.

A 3-adic number belongs to the completion of $\mathbb{Q}$ for the p-adic absolute value with prime three. Its nonzero elements have the form $3^{k}u$, with $k\in \mathbb{Z}$ and $u\in {\mathbb{Z}}_3^{\times }$. The cube-class group of the 3-adic numbers is generated by the classes of $3$ and $4$.

For a prime $p$, the $p$-adic absolute value on $\mathbb{Q}$ is $|x{|}_p=p^{-v_p(x)}$.

Every nontrivial absolute value on $\mathbb{Q}$ is equivalent either to the usual absolute value or to one $p$-adic absolute value.

The nontrivial non-Archimedean absolute values on a number field $K$, up to equivalence, are indexed by the nonzero prime ideals $\mathfrak{p}$ of its ring of integers. The value associated with $\mathfrak{p}$ is a fixed exponential of the discrete valuation $v_{\mathfrak{p}}$.

A local field is a nondiscrete locally compact topological field. A non-Archimedean local field is a finite extension of ${\mathbb{Q}}_p$ or of ${\mathbb{F}}_p((t))$.

A p-adic field is a finite extension of the P-adic number field ${\mathbb{Q}}_p$, equipped with its extended p-adic absolute value. Conversely every characteristic-zero, nontrivially valued, non-Archimedean locally compact field is such an extension. A finite quotient $\mathcal{O}/p\mathcal{O}$ supplies finitely many generators of $\mathcal{O}$ over ${\mathbb{Z}}_p$ by repeated reduction and completeness.

An integral lattice in a finite-dimensional ${\mathbb{Q}}_p$-vector space is a free ${\mathbb{Z}}_p$-submodule of full rank. Any two such lattices are commensurable: their intersection has finite index in each, because a sufficiently large power of $p$ multiplies either into the other. A finite group acting on the space admits invariant lattices by taking a sum of translates. This allows Herbrand quotient comparisons via finite quotient modules.

The degree of the residue field extension induced by a finite extension of complete discretely valued fields. Along with the ramification index, it satisfies $[L:K]=ef$. An unramified extension additionally has separable residue extension; a totally ramified extension has residue degree one.

A complete non-Archimedean valued field is locally compact exactly when its valuation is discrete and its residue field is finite. In that case its valuation ring is the inverse limit of the finite rings $\mathcal{O}/{\mathfrak{m}}^n$ and is compact.

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.

In a locally compact nontrivially valued field, the valuation ring is compact. A Cauchy sequence has a tail inside a translate of this compact set, hence a convergent subsequence. The Cauchy property forces the full sequence to converge.

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.

An infinite field with the trivial absolute value on a field is discrete, hence locally compact, but its residue field is the original infinite field. Thus local compactness forces a finite residue field and a discrete nontrivial value group only when the valuation is assumed nontrivial.

If a non-Archimedean local field $K$ has uniformizer $\pi$, valuation ring ${\mathcal{O}}_K$, maximal ideal ${\mathfrak{m}}_K$ and residue field ${\mathbb{F}}_q$, thenThe first factor records the discrete valuation, the Teichmuller representatives split reduction ${\mathcal{O}}_K^{\times }\to {\mathbb{F}}_q^{\times }$, and $1+{\mathfrak{m}}_K$ is the group of principal units.

If $K/{\mathbb{Q}}_p$ is finite of degree $d$, then as a topological abelian groupwhere ${\mu }_{p^a}(K)$ is the finite group of all roots of unity in $K$ whose orders are prime powers dividing a power of $p$.

For a complete discrete valuation ring $R$, a simple root of a polynomial modulo the maximal ideal lifts uniquely to a root in $R$.

Suppose a Non-Archimedean absolute value on $K$ has a unique extension of an absolute value to every finite extension. Factor a monic $g\in {\mathcal{O}}_K[X]$ into monic irreducible factors over $K$. All factors have integral coefficients because their roots have absolute value at most one. By pure-power reduction of a monic irreducible polynomial, each factor reduces to a power of a single irreducible polynomial. If $\overline{g}={\overline{g}}_1{\overline{g}}_2$ is a coprime monic factorization, assign each whole irreducible factor of $g$, with its multiplicity, to the unique side containing its residue factor. The resulting monic $g_1,g_2\in {\mathcal{O}}_K[X]$ satisfy $g=g_1{g}_2$ and reduce to the prescribed factors. Completeness of $K$ is not required for this proof.

For a non-Archimedean local field $K$, an element of $K^{\times }$ is an $m$th power for infinitely many positive integers $m$ exactly when it belongs to ${\mathcal{O}}_K^{\times }$. Valuations prove necessity; for sufficiency, choose infinitely many $m$ coprime to the residue characteristic and to $|k^{\times }|$, for which exponentiation by $m$ is an automorphism of the unit group.

The polynomial $X^4+9X^2-2$ has three irreducible factors over ${\mathbb{Q}}_2$, of degrees one, one, and two. Its quadratic in $X^2$ has one unit root congruent to one modulo eight, hence a square, and one root of valuation one, whose associated quadratic factor is Eisenstein.

Let $K$ be complete for a discrete valuation $v$. If $f\in {\mathcal{O}}_K[X]$ andthen Newton iteration over a valued field converges to a root $\alpha \in {\mathcal{O}}_K$ with $v(\alpha -a)>v(f^{\prime }(a))$.

Starting from $a_0$, Newton iteration sets $a_{n+1}=a_n-f(a_n)/f^{\prime }(a_n)$. Under the strong form of Hensel lemma inequality, the correction valuations grow at least geometrically, so $(a_n)$ is a Cauchy sequence converging to a root.

Let $K$ be complete, let $\alpha$ be separable over $K$, and let $\beta$ be algebraic over $K$. If $\beta$ is closer to $\alpha$ than any other $K$-conjugate of $\alpha$, then

A p-adic field has only finitely many extensions of a fixed degree up to isomorphism. There are finitely many possible residue degrees and a unique unramified extension of each such degree. Over each maximal unramified subfield, a totally ramified extension is generated by a root of an Eisenstein polynomial. The coefficient space of these polynomials of fixed degree is compact. Hensel lemma and Krasner's lemma show that sufficiently close polynomials generate isomorphic extensions, giving a finite cover by neighborhoods of a constant extension type.

The roots of a separable polynomial over a complete non-Archimedean field vary continuously with its coefficients after the roots are suitably matched. In particular, a sufficiently close coefficientwise approximation has one root in each sufficiently small disjoint ball around an original root.

Every finite extension of ${\mathbb{Q}}_p$ is isomorphic to the completion $F_{\mathfrak{p}}$ of some number field $F$ at a prime $\mathfrak{p}$ above $p$. Approximate a primitive element's minimal polynomial by a polynomial over $\mathbb{Q}$ and use Krasner's lemma to preserve the generated local field.

The completion ${\mathbb{C}}_p$ of an algebraic closure of ${\mathbb{Q}}_p$ is an algebraically closed field. Approximate a polynomial over ${\mathbb{C}}_p$ by one over ${\overline{\mathbb{Q}}}_p$, use continuity of roots over a non-Archimedean field to obtain a nearby algebraic root, and then use Krasner's lemma to show that the original root already belongs to ${\mathbb{C}}_p$.

Over a complete local ring, a power series whose first unit coefficient has index $d$ is a unit times a unique monic distinguished polynomial of degree $d$. Consequently its sufficiently small zeros, counted with multiplicity, are the zeros of that polynomial.

A distinguished polynomial over ${\mathbb{Z}}_p$ is monic and has every non-leading coefficient divisible by $p$. The Weierstrass preparation theorem expresses any nonzero element of ${\mathbb{Z}}_p[[T]]$ as a power of $p$ times a unit times a distinguished polynomial.

The Laurent series field $k((t))$ consists of formal series ${\sum }_{n\ge N}{a}_n{t}^n$ with finitely many negative-degree terms. Its $t$-adic valuation is the least exponent with nonzero coefficient.

For an irreducible polynomial $h\in {\mathbb{F}}_p[t]$ of degree $d$, the $h$-adic completion of ${\mathbb{F}}_p(t)$ is isomorphic to ${\mathbb{F}}_{p^d}((T))$. Its residue field is ${\mathbb{F}}_p[t]/(h)$ and $h$ is a uniformizer.

The $p$-adic exponential $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ converges near zero.

On ${\mathbb{C}}_p$, the power series $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ has radius of convergenceIndeed, Legendre formula gives $v_p(n!)/n\to 1/(p-1)$, and the Cauchy-Hadamard theorem gives the stated radius.

The $p$-adic logarithm is the convergent series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ on a sufficiently small group of principal units.
It is locally inverse to the p-adic exponential function.

For a mixed-characteristic local field with residue characteristic $\ell$, the p-adic logarithm and p-adic exponential function are mutually inverse continuous group homomorphisms on these domains. Bounds on the valuations of the denominators show that higher terms have greater valuation than the linear term. Dividing the logarithm by ${\pi }^r$ identifies the multiplicative higher principal-unit group with the additive integer ring; in particular it is torsion-free.