Intereting Posts

measure of a countable union of sets
Prob. 11, Chap. 4 in Baby Rudin: uniformly continuous extension from a dense subset to the entire space
for a $3 \times 3$ matrix A ,value of $ A^{50} $ is
block matrices problem
Projection and Pseudocontraction on Hilbert space
How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history?
The Ellipse Problem – finding an ellipse inside a triangle
Two congruent segments does have the same length?
Generating function for vertices distance from the root in a planar tree
Hilbert spaces and unique extensions of linear functions.
The total ring of fractions of a reduced Noetherian ring is a direct product of fields
Existence of submersions from spheres into spheres
Calculate variance from a stream of sample values
When does it make sense to define a generator of a set system?
Unique Decomposition of Primes in Sums Of Higher Powers than $2$

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.

- What lies beyond the Sedenions
- What's the difference between isomorphism and homeomorphism?
- Existence of an operation $\cdot$ such that $(a*(b*c))=(a\cdot b)*c$
- Idealizer of one-sided ideal
- Why $x^{p^n}-x+1$ is irreducible in ${\mathbb{F}_p}$ only when $n=1$ or $n=p=2$
- Sylow questions on $GL_2(\mathbb F_3)$.

- Difference between i and -i
- Finding inverse in non-commutative ring
- Which polynomials with binary coefficients evaluate only to 0 or 1 over an extension field?
- Localisation and prime ideals
- Is it true that $p\mid $?
- Structure theorem for finitely generated abelian groups
- Finite Abelian groups with the same number of elements for all orders are isomorphic
- Let $p$ be a prime, then does $p^{\alpha} \mid |G| \Longrightarrow p \mid Aut(G)$?
- Could $X^3 + X + 1$ have a root in $E?$
- counting the number of elements in a conjugacy class of $S_n$

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

- If $f^2$ is Riemann Integrable is $f$ always Riemann Integrable?
- Effect of differentiation on function growth rate
- Prove $\sum_{n=0}^{\infty}{2^n(n^2-n\pi+1)(n^2+n-1)\over (2n+1)(2n+3){2n\choose n}}=1$
- Every integer vector in $\mathbb R^n$ with integer length is part of an orthogonal basis of $\mathbb R^n$
- Subgroups of the Symmetric Group
- How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?
- Show that equation has no solution in $(0,2\pi)$
- Prove optimal solution to dual is not unique if optimal solution to the primal is degenerate and unique.
- Significance of starting the Fibonacci sequence with 0, 1…
- Solution of $ax=a^x$
- Not sure how to go about solving this integral
- Non-finitely generated, non-projective flat module, over a polynomial ring
- What comes after $\cos(\tfrac{2\pi}{7})^{1/3}+\cos(\tfrac{4\pi}{7})^{1/3}+\cos(\tfrac{6\pi}{7})^{1/3}$?
- Various proofs of Hardy's inequality
- Which step in this process allows me to erroneously conclude that $i = 1$