Intereting Posts

Zeros of a holomorphic function
Wallis Product for $n = \tfrac{1}{2}$ From $n! = \Pi_{k=1}^\infty (\frac{k+1}{k})^n\frac{k}{k+n} $
Summing over General Functions of Primes and an Application to Prime $\zeta$ Function
Determinant of circulant matrix
Why $(1-\zeta)$ unit where $\zeta$ is a primitive nth and n divisible by two primes
How does Wolfram Alpha come up with the substitution $x = 2\sin u$? Integration/Analysis
Distance between triangle's centroid and incenter, given coordinates of vertices
What does the big intersection or union sign of a set mean?
Prove that $\log _5 7 < \sqrt 2.$
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
Krull dimension of the injective hull of residue field
Name for Cayley graph of a semigroups?
Meaning of $f:\to\mathbb R$
Suppose $p$ is a prime number and $a$ is an integer. Show that if $p \mid a^n$, then $p^n \mid a^n$ for any $n \geq 1$?
Finding the area under the curve $y=3-3\cos(t),x=3t-3\sin(t)$

A commutative ring $F$ is a field iff $F[x]$ is a Principal Ideal Domain.

I have done the part that if $F$ is a field then $F[x]$ is a PID using the division algorithm and contradicting the minimality of degree of a polynomial.

But I am facing difficulty to do the other part.

- Groups of units of $\mathbb{Z}\left$
- an example for $\pi$-quasinormal subgroup but is not normal
- Direct sum of non-zero ideals over an integral domain
- Galois groups of polynomials and explicit equations for the roots
- How to classify one-dimensional F-algebras?
- Is the image of a tensor product equal to the tensor product of the images?

- Finite fields and primitive elements
- Quotient ring of Gaussian integers $\mathbb{Z}/(a+bi)$ when $a$ and $b$ are NOT coprime
- Zero image of an element in the direct limit of modules
- How do you prove the ideal $I= (X^2, XY)$ has infinitely many distinct irredundant primary decompositions?
- An integer square matrix of prime order has size at least $(p-1)\times (p-1)$
- When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?
- Ring of integers is a PID but not a Euclidean domain
- A subgroup of $\operatorname{GL}_2(\mathbb{Z}_3)$
- Non-UFD integral domain such that prime is equivalent to irreducible?
- Prove that $p$ is prime in $\mathbb{Z}$ if and only if $x^2+3$ is irreducible in $\mathbb{F}_p$.

Suppose $k[X]$ is a PID. Prove that $(X)$ is a maximal ideal and then note $k\simeq k[X]/(X)$.

- Equivalent metrics using open balls
- General solution for the Eikonal equation $| \nabla u|^2=1$
- Integration of $e^{ax}\cos bx$ and $e^{ax}\sin bx$
- Hydrostatic pressure on an equilateral triangle
- A group such that $a^m b^m = b^m a^m$ and $a^n b^n = b^n a^n$ ($m$, $n$ coprime) is abelian?
- Taking limits on each term in inequality invalid?
- Complex integral
- Prove for all sequences $\{a_n\}$ and $\{b_n\}$, if $\lim a_n = a$ and $\lim b_n = b$, then $\lim a_n + b_n = a+b$ entirely in first-order logic
- A certain inverse limit
- Status of the classification of non-finitely generated abelian groups.
- The action of a Galois group on a prime ideal in a Dedekind domain
- Show that, if $f:A\to B$ is a function, with $A$ and $B$ being finite sets, and $|A|=|B|$, then $f$ is one to one iff $f$ is onto.
- Find all the values of $(1+i)^{(1-i)}$
- How to evaluate $\int _{-\infty }^{\infty }\!{\frac {\cos \left( x \right) }{{x}^{4}+1}}{dx}$
- derivative with respect to constant.