For any continuous function $f:\mathbb{Z}_p \to \mathbb{Q}_p$, Mahler's theorem provides us with a relatively explicit series of polynomials converging uniformly to $f$. Is there any analogue for other non archimedean fields? In particular, what about the completion $\mathbb{C}_p$ of the algebraic closure of $\mathbb{Q}_p$?
2026-02-23 01:21:43.1771809703
Analog of Mahler's theorem over other non archimedean fields
36 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail AtRelated Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
Related Questions in ANALYSIS
- Analytical solution of a nonlinear ordinary differential equation
- Finding radius of convergence $\sum _{n=0}^{}(2+(-1)^n)^nz^n$
- Show that $d:\mathbb{C}\times\mathbb{C}\rightarrow[0,\infty[$ is a metric on $\mathbb{C}$.
- conformal mapping and rational function
- What are the functions satisfying $f\left(2\sum_{i=0}^{\infty}\frac{a_i}{3^i}\right)=\sum_{i=0}^{\infty}\frac{a_i}{2^i}$
- Proving whether function-series $f_n(x) = \frac{(-1)^nx}n$
- Elementary question on continuity and locally square integrability of a function
- Proving smoothness for a sequence of functions.
- How to prove that $E_P(\frac{dQ}{dP}|\mathcal{G})$ is not equal to $0$
- Integral of ratio of polynomial
Related Questions in NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- Show that $(x,y,z)$ is a primitive Pythagorean triple then either $x$ or $y$ is divisible by $3$.
- About polynomial value being perfect power.
- Name of Theorem for Coloring of $\{1, \dots, n\}$
- Reciprocal-totient function, in term of the totient function?
- What is the smallest integer $N>2$, such that $x^5+y^5 = N$ has a rational solution?
- Integer from base 10 to base 2
- How do I show that any natural number of this expression is a natural linear combination?
- Counting the number of solutions of the congruence $x^k\equiv h$ (mod q)
Related Questions in P-ADIC-NUMBER-THEORY
- Can $\mathbb{Z}_2$ be constructed as the closure of $4\mathbb{Z}+1$?
- Number of points in reduction of a p-adic analytic manifold.
- How do I translate functions on the Prufer 2-group between functions on the $2^n$ roots of unity and the dyadic fractions modulo 1?
- Hensel Lemma and cyclotomic polynomial
- orbit representatives for the group of unipotent matrix acting on the set of skew-symmetric matrices
- Homomorphic images of $p$-adic integers
- Criteria for a cubic polynomial in $\Bbb Q[x]$ to split completely over $\Bbb Q_p$
- What do the elements of the affinoid algebra $A=K\langle x, y\rangle/(y-\pi x)$ look like?
- Find $\frac{a}{b} \in \mathbb{Q}$ such that $ |\,\frac{a}{b} - \sqrt{2}|_3 < \epsilon $
- Hilbert symbol Definitons
Related Questions in NONARCHIMEDIAN-ANALYSIS
- Example of a Map of Banach Spaces over a Non-Archimedian Field with Non-Closed Image
- Consider the p-adic field $ \ \mathbb{Q}_p \ $ . Define $ \operatorname{ord}_p(x) \ $ to be the p-adic valuation of $ \ x \ $
- How to write disc of convergence in $ (1) $ in the following form $ |x|_p < K $
- Does the above non-Archimedean but ordered field satisfy Nested interval property?
- Structure of units in unramified extension of $\mathbb{Q}_2$
- $p$-adic power series and its maximum in the unit ball
- An equality of polynomials
- Reference Request: Manifold theory when $\mathbb{R}$ is replaced by a complete ordered field
- Why does $f(x_n)\rightarrow 0$ imply that a subsequence converges to zero of $f$?
- unit ball in $\mathbb C_p$ as an inverse limit
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?