All Questions
12
questions
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34
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Meaning of irreducible polynomial that is factor of $P$
I am reading the following lemma about polynomials:
Suppose that $x$ is a root of a polynomial $P$, $\pmod p$. Then the
irreducible polynomial $(T - x)$ is a factor of $P$
I am not sure I understand ...
- 1,609
1
vote
1
answer
52
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Why is $x^4+x^2+1$ over $đť”˝_2$ a reducible polynomial? What do I misunderstand?
I don't quite understand when a polynomial is irreducible and when it's not.
Take $x^2 +1$ over $đť”˝_3$.
As far as I know, I have to do the following:
0 1 2 using $x \in đť”˝_3$
1 2 2 using $p(x)$
I ...
- 931
4
votes
2
answers
472
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Is $x^6 + 108$ irreducible over $\mathbb{Q}$?
I'm trying to determine whether or not $x^6 + 108$ is irreducible over $\mathbb{Q}$. Is there an easy way to determine this ? I tried Eisenstein's Criterion directly, and with the substitutions $x \...
- 41
1
vote
1
answer
94
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Find prime fields over which a polynomial has roots.
Suppose we have a polynomial
$$h(x) = a_n x^n + \dots + a_1 x + 1$$
Given the values $a_1,\ldots,a_n$, how to determine whether there exists such prime $p$ that $h(x)$ has roots over the field $\...
- 3,212
2
votes
2
answers
44
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Working with polynomials in $\mathbb{Z}_{2}[X]$, finding roots and splitting fields
I'm a beginner in the area and yet can't see how to work with a polynomial in other fields diferents from $\mathbb{Q}$.
I have the polynomial $f(X)=X^5 -X^2+1 \in \mathbb{Z}_{2}[X]$ and must prove ...
- 1,403
6
votes
1
answer
590
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Does every polynomial over a finite field have a square root modulo an irreducible polynomial?
Given a polynomial $p \in \operatorname{GF}(2^m)[x]$ and an irreducible polynomial $g \in GF(2^m)[x]$, is there a $d \in \operatorname{GF}(2^m)[x]$ such that $d^2(x) = p \pmod{g(x)}$?
In other words, ...
- 75
1
vote
1
answer
126
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Divisors of polynomials in $\mathbb{Z}_p$ for $p$ prime
Let $f \in \mathbb{Z}[x]$ be a non-constant polynomial where $p$ is a prime that is not a divisor of the leading coefficient of $f$.
Assume that $f = g_{1}g_{2},\ldots, g_{k}$ is a factorization of $...
user100463
5
votes
1
answer
3k
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$f$ irreducible over $\mathbb{Z}_{p}$ implies $f$ is irreducible over $\mathbb{Q}$
Let $f \in \mathbb{Z}[x]$ be a non-constant polynomial and let $p$ be a prime number which is not a divisor of the leading coefficient of $f$. I need to prove that if $f$ is irreducible over $\mathbb{...
user100463
2
votes
1
answer
3k
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Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$.
Find all the Zeros and their multiplicities of $f(x)=x^5 +4x^4 +4x^3 -x^2-4x +1$ over $\Bbb Z_5$.
Firstly,I've found the zeros of $f(x)$,just by simply substituting the elements of $\Bbb Z_5=\{0,1,2,...
- 3,569
2
votes
1
answer
41
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Stuck with the statement: $t^4+2$ in $\mathbb{Z}_5$ gives rise to...
This is from Ian Stewart's book on Galois theory, I am looking at irreducible polynomials.
It talks of irreducibility over mod. It takes as an example, $f(t)=t^4+15t^3+7$ over integers, and asks us ...
- 3,299
2
votes
1
answer
65
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Roots of $x^2+1$ over $\mathbb{Z}_7$?
Prove that $x^2 + 1$ is irreducible in the ring of polynomials $\Bbb Z _7 [x]$ over the field $\Bbb Z _7$.
Is it enough to show that no single element of $\mathbb{Z}_7$ squared is equal to $-1 \pmod ...
- 991
2
votes
2
answers
914
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How can I prove the polynomial f is irreducible
We have $f\in \mathbb{Z}_{3}\left[X\right],\:\:f=x^3+2x^2+a,\:\:a\in \mathbb{Z}_{3}$ and we need to find $a$ for which polynomial $f$ is irreducible.
I looked on google but I don't understand very ...
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