Intereting Posts

Divergence of $\sum\frac{\cos(\sqrt{n}x)}{\sqrt{n}}$
Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}$
Parametric QR factorization: $\mathbf{D}(\alpha)\mathbf{V}^*$ a diagonal times a constant unitary matrix
Sum of the series formula
Good problem book on Abstract Algebra
When are commutative, finite-dimensional complex algebras isomorphic?
How to solve the following integral?
Express this sum of radicals as an integer?
factorize $x^5+ax^3+bx^2+cx+d$ if $d^2+cb^2=abd$
Do all vectors have direction and magnitude?
if $f$ is differentiable at a point $x$, is $f$ also necessary lipshitz-continuous at $x$?
How to show the following process is a local martingale but not a martingale?
Is the product of a Cesaro summable sequence of $0$s and $1$s Cesaro summable?
Supremum of a Set of Integers
Can someone explain how the Schreier-Sims Algorithms works on a permutation group with a simple example?

Consider the polynomial ring $\Bbb F_m[x]$, and two polynomials $f,g $ in $\Bbb F_m[x]$.

Is there any necessary and sufficient conditions for $f,g$ such that

$\Bbb F_m[x]/(f) \cong \Bbb F_m[x]/(g)$?

Or if there are two particular $f,g$ in $\Bbb F_m[x]$, how to check that the quotient rings generated by $f,g$ are isomorphic? I have this question when I encounter the following question:

- Number of elements which are cubes/higher powers in a finite field.
- Compatibility of direct product and quotient in group theory
- What do we call collections of subsets of a monoid that satisfy these axioms?
- Why don't we have an isomorphism between $R$ and $ R]$?
- Is the splitting field equal to the quotient $k/(f(x))$ for finite fields?
- How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

Are $\Bbb F_3[x]/(x^3+x^2+x+1), \Bbb F_3[x]/(x^3-x^2+x-1) $

isomorphic?

I could not see a suitable isomorphism so I assume they are not isomorphic, yet cannot come to a contradiction so far.

Any help for the two problems is appreciated, thanks.

- How to determine the matrix of adjoint representation of Lie algebra?
- Proofs for formal power series
- Example of non-trivial number field
- Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
- Cyclotomic polynomials explicitly solvable??
- $A_4$ has no subgroup of order $6$?
- Example of non-flat modules
- The Jacobson Radical of a Matrix Algebra
- A book for abstract algebra with high school level
- Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)

If $f$ and $g$ are irreducible, then the two quotient rings are fields and so are isomorphic iff $f$ and $g$ have the same degree, because there is only one finite field of each possible cardinality.

This argument can be generalized for squarefree polynomials, in which case the quotient rings are products of finite fields.

- Conjectures that have been disproved with extremely large counterexamples?
- Delta function integrated from zero
- How to factorize polynomial in GF(2)?
- Why would the reflections of the orthocentre lie on the circumcircle?
- Showing that the diagonal of $G \times G$ is maximal, where $G$ is simple
- Is reducing a matrix to row echelon form useful at all?
- Solving contour integral
- Is There Something Called a Weighted Median?
- Integral $\int_0^1 \log \Gamma(x)\cos (2\pi n x)\, dx=\frac{1}{4n}$
- Compute $e^{e^z}$, where z is a complex number
- Proving $2^{2n}-1$ is divisible by $3$ for $n\ge 1$
- Sum of squares diophantine equation
- Abstract algebra book recommendations for beginners.
- Does $\pi$ have infinitely many prime prefixes?
- Product of compact and closed in topological group is closed