All Questions
5
questions
0
votes
1
answer
69
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Reducible/Irreducible Polynomials in Ring Theory
I have this following exercise I've been trying to solve for a while now.
We are supposed to study the irreducibility of the polynomial $A=X^4 +1$ in $\mathbb{Z}[X]$ and in $\mathbb{Z}/p \mathbb{Z}$ ...
- 11
2
votes
2
answers
54
views
Units, Primes and Irreducibles
How do you find the units, irreducible elements and prime elements for $\mathbb{C}[𝑥]$, $\mathbb{R}[𝑥]$, $\mathbb{Q}[𝑥]$?
Thank you.
- 51
2
votes
0
answers
39
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Irreducibility of polynomials in $F[X]$
Let $f = 3X^4 + 2X^3Y + X^2Y^2 + 3XY^3 + Y^4$. Let $R =\mathbb{Z}[Y]$, $F = Frac[R]$.
I want to know whether or not $f$ is irreducible in $F[X]$. I've tried using the Eisentstein criterion, but I ...
- 2,482
-1
votes
1
answer
596
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An Analogue of CRT for Polynomial Ring
How can we determine the number of factors by CRT?
For example,
$$\frac{\mathbb{Z}_5[X]}{X^2+1}\cong \frac{\mathbb{Z}_5[X]}{X+2}\times \frac{\mathbb{Z}_5[X]}{X+3},$$
so $\frac{\mathbb{Z}_5[X]}{X^2+...
- 829
0
votes
1
answer
175
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Prove that the elements $2x$ and $x^2$ have no LCM in the ring of integral polynomials with even coefficient of $x$
Let $A$ be the subring of $\Bbb Z[x]$ consisting of all polynomials with even coefficient of $x$. Prove that the elements $2x$ and $x^2$ have no lowest common multiple.
Hints please!
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