Intereting Posts

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Closed form for $\sum_{n=1}^{\infty}\frac{1}{\sinh^2\!\pi n}$ conjectured
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Is there exist a homemoorphism between either pair of $(0,1),(0,1],$
Show that the eigenvalues of a unitary matrix have modulus $1$
Spectrum of the derivative operator
Proof for an equality involving square roots
Combination problem distributing
On finite 2-groups that whose center is not cyclic
How to read a book in mathematics?
How to find a general sum formula for the series: 5+55+555+5555+…?
Non-isomorphic graphs with four total vertices, arranged by size

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:

- Let $L/K$ be a Galois extension with $Gal(L/K)=A_{4}$. Prove that there is no intermediate subfield $M$ of $L/K$ such that $=2$.
- Group of even order contains an element of order 2
- Cardinality of a minimal generating set is the cardinality of a basis
- Multiplicative group $(\mathbb R^*, ×)$ is group but $(\mathbb R, ×)$ is not group, why?
- A non-UFD such that $a^2 \mid b^2$ does not lead to $a\mid b$
- units of a ring of integers

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.

- Presentation of group equal to trivial group
- The degree of $\sqrt{2} + \sqrt{5}$ over $\mathbb Q$
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- The primitive spectrum of a unital ring
- Zero image of an element in the direct limit of modules
- Showing a Ring of endomorphisms is isomorphic to a Ring
- What is so special about Higman's Lemma?
- Semisimplicity is equivalent to each simple left module is projective?
- Is a topological group action continuous if and only if all the stabilizers are open?
- Galois theory reference request

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.

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- Prove or disprove that the given expression is “always” positive
- “This statement is false.”
- Why is the collection of all algebraic extensions of F not a set?
- Let $X$ be the number of aces and $Y$ be the number of spades. Show that $X$, $Y$ are uncorrelated.
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- Non-free modules, with a free direct sum
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- Applications of complex numbers to solve non-complex problems
- Exercise on $\dim \ker T^i$, with $T^4=0$