A local field is a field that is locally compact and complete w.r.t. absolute value.
Does there exist a local field $K$ which is compact?
2026-03-25 15:45:25.1774453525
Examples of local fields
189 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
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 ELEMENTARY-NUMBER-THEORY
- Maximum number of guaranteed coins to get in a "30 coins in 3 boxes" puzzle
- Interesting number theoretical game
- How do I show that if $\boldsymbol{a_1 a_2 a_3\cdots a_n \mid k}$ then each variable divides $\boldsymbol k $?
- Using only the digits 2,3,9, how many six-digit numbers can be formed which are divisible by 6?
- Algebra Proof including relative primes.
- 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)
- algebraic integers of $x^4 -10x^2 +1$
- What exactly is the definition of Carmichael numbers?
- Number of divisors 888,888.
Related Questions in FIELD-THEORY
- Square classes of a real closed field
- Question about existence of Galois extension
- Proving addition is associative in $\mathbb{R}$
- Two minor questions about a transcendental number over $\Bbb Q$
- Is it possible for an infinite field that does not contain a subfield isomorphic to $\Bbb Q$?
- Proving that the fraction field of a $k[x,y]/(f)$ is isomorphic to $k(t)$
- Finding a generator of GF(16)*
- Operator notation for arbitrary fields
- Studying the $F[x]/\langle p(x)\rangle$ when $p(x)$ is any degree.
- Proof of normal basis theorem for finite fields
Related Questions in ABSOLUTE-CONVERGENCE
- Does one-sided derivative of real power series at edge of domain of convergence
- Every rearrangement of an absolutely convergent series converges to the same sum (Rudin)
- Prove $\int_{\pi}^{\infty}\frac{\cos(x)}{x}dx$ is convergent
- Conditional convergent improper Riemann integral vs. Lebesgue Integral
- Pointwise, uniform and absolute convergence of function-series $\sum_{n=1}^\infty f_n$ with $f_n=(-1)^n\frac{x}{n}$
- Does it make mathematical sense to do an absolute convergence test if the original series diverges?
- Prove absolute convergence given the following inequality
- Proving a space is not complete by finding an absolutely convergent series
- Convergence and absolute convergence of sums
- Absolute convergence of $\sum_{n=1}^{\infty}z_n^2$
Related Questions in LOCAL-FIELD
- What is meant by a category of unramified extension of $K$ (a local field)?
- Given a local field is the maximal unramifield extension always finite?
- $(K^*)$ in $K^*$
- How do we know that reducing $E/K$ commutes with the addition law for $K$ local field
- How is $\operatorname{Gal}(K^{nr}/K)$ isomorphic to $\operatorname{Gal}(\bar{k}/k)$?
- Extending a valuation of a local field
- On the Galois group of the maximal $p$-abelian $p$-ramified extension of a number field
- Finite extension of $K$,a finite extension of $\mathbb{Q}_p$, that is Galois?
- Why is $E[2]$ of $E: y^2 = x^3 - p$ over $\mathbb{Q}_p$ ramified?
- Are there any $\mathbb{Q}_p$ which contains the third root of unity?
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?
Let $R$ be a unital associative ring, endowed with a topological ring topology (say, addition and multiplication are continuous). If $R$ is compact then the set of left invertible elements in $R$ is closed (indeed, it is the first projection of the closed subset $\{(x,y):xy=1\}$. In particular, if $R\neq\{0_R\}$, then $0_R$ does not belong to the closure of the set of left-invertible elements. In particular, the only (Hausdorff) compact topological fields (or skew-fields) are finite.