All Questions
10
questions
0
votes
1
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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
0
answers
92
views
Non-constant polynomial over an integral domain without any irreducible factors.
Let $R$ be an integral domain. I am trying to find a $f \in R[x]$, such that $\deg(f) \geq 1$, and $f$ does not have any irreducible factors in $R[x]$.
Does such $f$ exist?
Though I haven't been able ...
- 69
1
vote
2
answers
136
views
Justifying the representation of a monic polynomial over a UFD
Let $D$ be a UFD and $f(x) \in D[x]$ be monic. The book I'm reading from claims that $$f(x) = p_1(x)^{e_1} \cdots p_n(x)^{e_n}$$
where $p_i(x)$ are distinct, irreducible, and monic, and $e_i >0$.
...
- 4,039
0
votes
1
answer
359
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Can the linear polynomials be totally ordered?
At the start of this document http://www.math.lsa.umich.edu/~lagarias/575chomework/p-adic-chap5.pdf Lagarias draws the analogy between writing any integer in base $p$, and writing any polynomial in ...
- 9,299
1
vote
1
answer
94
views
Ideal decomposition: when does $(I,xy)=(I,x) \cap (I,y)$?
Let I be an ideal in a commutative ring with 1 A. Let $x,y \in A$. When does the formula $(I,xy)=(I,x) \cap (I,y)$ hold?
One may prove this is true when $x \neq y$ are irreducibles and $(x,y) \cap I= ...
- 103
0
votes
1
answer
110
views
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$? [closed]
The question is as in the title:
Is every polynomial with integer coefficients prime in $\mathbb{Z}[x]$ also prime in $\mathbb{Q}[x]$?
- 157
1
vote
1
answer
369
views
If $f \mid h, g\mid h$ and $f,g$ are relatively prime, then $fg\mid h$?
Let $f,g,h \in\mathbb{F}[x]$, with $f$ and $g$ are relatively prime. If $f\mid h$ and $g\mid h$, prove that $fg\mid h$.
What I've done so far: Experimenting with natural numbers, I suspect that $h=fg\...
- 7,558
7
votes
1
answer
211
views
Irreducibility of an integer polynomial with exponents in linear sequence?
Let $b$ and $n$ be two positive integers. Is there are a general result which tell us when the polynomial $$1+x^{b}+x^{2b}+x^{3b}+\cdots+x^{nb}$$
is irreducible over the integers?
I know that $$1+x+...
user123641
2
votes
3
answers
236
views
Number of solutions of polynomials in a field
Consider the polynomial $x^2+x=0$ over $\mathbb Z/n\mathbb Z$
a)Find an n such that the equation has at least 4 solutions
b)Find an n such that the equation has at least 8 solutions
My idea is to ...
- 171
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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