We know $Q$ with any of $p$ aidic norms is not complete for all primes including infinity. Now my question is can we make same conclusion for any countably infinite fields ? i.e does there exist a countably infinite normed linear field which is complete ?
2026-03-26 13:49:39.1774532979
does there exist a countably infinite normed field which is complete?
84 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in P-ADIC-NUMBER-THEORY
- How does one define an inner product on the space $V=\mathbb{Q}_p^n$?
- 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 $
Related Questions in VALUATION-THEORY
- Does every valuation ring arise out of a valuation?
- Calculating the residue field of $\mathbb{F}_q(t)$ with respect to the valuation $-\deg$
- Different discrete valuation rings inside the same fraction field
- Extensions of maximally complete fields
- State Price Density Integration - Related to Ito's lemma
- Valuation on the field of rational functions
- Is every algebraically closed subfield of $\mathbb C[[X]]$ contained in $\mathbb C$?
- Working out an inequality in the proof of Ostrowski's Theorem
- The image of a valuation is dense in $\mathbb{R}$
- Kernel of the map $D^2+aD+b : k[[X]] \to k[[X]]$ , where $D :k[[X]] \to k[[X]]$ is the usual derivative map
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?
Such a field is a topological group, hence homogeneous. This answer shows such a space must be discrete. Can you show normed fields cannot be discrete?