My professor gave an example of a free group as the field of rationals over one variable $\mathbb C(x)^*$ but I do not know how. could anyone explain this to me, please?
2026-03-29 04:48:48.1774759728
How is $\mathbb C(x)^*$ a free group?
53 Views Asked by user778657 https://math.techqa.club/user/user778657/detail At
1
There are 1 best solutions below
Related Questions in POLYNOMIALS
- Alternate basis for a subspace of $\mathcal P_3(\mathbb R)$?
- Integral Domain and Degree of Polynomials in $R[X]$
- Can $P^3 - Q^2$ have degree 1?
- System of equations with different exponents
- Can we find integers $x$ and $y$ such that $f,g,h$ are strictely positive integers
- Dividing a polynomial
- polynomial remainder theorem proof, is it legit?
- Polyomial function over ring GF(3)
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- $x^{2}(x−1)^{2}(x^2+1)+y^2$ is irreducible over $\mathbb{C}[x,y].$
Related Questions in EXAMPLES-COUNTEREXAMPLES
- A congruence with the Euler's totient function and sum of divisors function
- Seeking an example of Schwartz function $f$ such that $ \int_{\bf R}\left|\frac{f(x-y)}{y}\right|\ dy=\infty$
- Inner Product Uniqueness
- Metric on a linear space is induced by norm if and only if the metric is homogeneous and translation invariant
- Why do I need boundedness for a a closed subset of $\mathbb{R}$ to have a maximum?
- A congruence with the Euler's totient function and number of divisors function
- Analysis Counterexamples
- A congruence involving Mersenne numbers
- If $\|\ f \|\ = \max_{|x|=1} |f(x)|$ then is $\|\ f \|\ \|\ f^{-1}\|\ = 1$ for all $f\in \mathcal{L}(\mathbb{R}^m,\mathbb{R}^n)$?
- Unbounded Feasible Region
Related Questions in FREE-GROUPS
- How to construct a group whose "size" grows between polynomially and exponentially.
- Help resolving this contradiction in descriptions of the fundamental groups of the figure eight and n-torus
- What is tricky about proving the Nielsen–Schreier theorem?
- Abelian Groups and Homomorphic Images of Free Abelian Groups
- Proof check, existence of free product
- determine if a subgroup of a free group is normal
- Bass-Serre tree of Isom($\mathbb{Z}$)
- Finitely Generated Free Group to Finitely Generated Free Monoid
- Crossed homomorphism/derivation on free group
- Existence of elementd of infinite order in finitely generated infinite group
Related Questions in UNIQUE-FACTORIZATION-DOMAINS
- Extension and restriction of involutions
- Why is this element irreducible?
- What is the correct notion of unique factorization in a ring?
- A question about unique factorization domain
- Is the union of UFD an UFD?
- Is $F_p^{l}[t]$ is a UFD
- etymology of smoothness
- $2=(1+i)(1-i)$ what does that imply in $\mathbb{Z}[i]$?
- Is there a way of proving that $\mathbb{Z}[i]$ and $\mathbb{Z}[\sqrt{-2}]$ are UFD s without showing that they are euclidian domains?
- What implies that $D[X]$ is an UFD?
Related Questions in POLYNOMIAL-RINGS
- Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.
- For what $k$ is $g_k\circ f_k$ invertible?
- Prove that the field $k(x)$ of rational functions over $k$ in the variable $x$ is not a finitely generated $k$-algebra.
- The 1-affine space is not isomorphic to the 1-affine space minus one point
- What are the coefficients of $x^2+2\in(\mathbb{Z}/\mathbb{Z}4)[x]?$
- Prove that $\mathbb{Z}_{5}[x]/(x^2+1)$ is isomorphic to $\mathbb{Z}_{5} \times \mathbb{Z}_{5}$.
- Polynomial ring over finite field - inverting a polynomial non-prime
- Descending Chain Condition
- notation in congruence relation
- Is the cardinality of the polynomial quotient ring $\mathbb{Z}_n [x] /f(x)$ always finite?
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?
You mean a free abelian group, a free group is completely different.
$\Bbb{C} (x)^\times/\Bbb{C}^\times$ (isomorphic to the quotients of monic polynomials) is a free abelian group, generated by the $x-a,a\in \Bbb{C}$.
$\Bbb{C} (x)^\times$ is not a free abelian group as $(-1)^2= 1$. Also for any fixed $z\in \Bbb{C}^\times$, there is a $z^{1/n}$ for all $n$.