Let $k$ be a field such that its group of units $k^\times$ is finitely generated ; then is it true that $k$ is finite ? I can only show that $\text{char}(k) >0$ . Please help . Thanks in advance
2026-03-29 12:05:41.1774785941
Field whose group of units is finitely generated
231 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in RING-THEORY
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- A commutative ring is prime if and only if it is a domain.
- Find gcd and invertible elements of a ring.
- Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.
- Prove that $Z[i]/(5)$ is not a field. Check proof?
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- Let $R$ be a simple ring having a minimal left ideal $L$. Then every simple $R$-module is isomorphic to $L$.
- A quotient of a polynomial ring
- Does a ring isomorphism between two $F$-algebras must be a $F$-linear transformation
- Prove that a ring of fractions is a local ring
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 ABELIAN-GROUPS
- How to construct a group whose "size" grows between polynomially and exponentially.
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- Invariant factor decomposition of quotient group of two subgroups of $\mathbb{Z}^n$.
- Computing Pontryagin Duals
- Determine the rank and the elementary divisors of each of the following groups.
- existence of subgroups of finite abelian groups
- Theorem of structure for abelian groups
- In the category of abelian groups the coequalizer $\text{Coker}(f, 0)$, $f: A \to B$ is simply $B/f(A)$.
- Commutator subgroup and simple groups
- Are there any interesting examples of functions on Abelian groups that are not homomorphisms?
Related Questions in FINITELY-GENERATED
- projective module which is a submodule of a finitely generated free module
- Ascending chain of proper submodules in a module all whose proper submodules are Noetherian
- Is a connected component a group?
- How to realize the character group as a Lie/algebraic/topological group?
- Finitely generated modules over noetherian rings
- Integral Elements form a Ring
- Module over integral domain, "Rank-nullity theorem", Exact Sequence
- Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian!
- Computing homology groups, algebra confusion
- Ideal Generated by Nilpotent Elements is a Nilpotent Ideal
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 would be a quotient of $R=\Bbb Z[X_1,\ldots,X_n]$. It is fairly well-known that all quotient fields of $R$ are finite. If you know the characteristic is $p$, then such a field is a quotient of $\Bbb F_p[X_1,\ldots,X_n]$ and so by the Nullstellensatz is a finite extension of $\Bbb F_p$.
(I omit the proof that the characteristic in nonzero; that is another application of the Nullstellensatz.)