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Tagged with polynomials irreducible-polynomials
1,520
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Irreducible polynomials in $\mathbb F_q[T]$
Let $q$ be a power of a prime $p$. Is there an infinite set $S$ of $\mathbb N$ such that for every $l\in S$, the polynomial $T^{q^l}-T-1$ is irreducible in $\mathbb F_q[T]$.
It looks like Artin-...
- 1,157
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Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$
How do I factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$?
I checked that the discriminant $D = 16 -4 = 12$ is not a square ($12^{14} = -1 \mod 29$) so this polynomial has no roots. Therefore it's ...
- 2,636
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Is $X^4-3X^2+2X+1$ irreducible over the rationals?
I want to check whether the following polynomials is irreducible over the rationals $f=X^4-3X^2+2X+1$. I think I found that it is irreducible but my solution is really complicated, quite long and I am ...
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Question about reduction mod p method for polynomials [duplicate]
I am a undergradute math student and I was traying to prove the polynomial $p(x)= x^4-10x^2+1$ is irreducible in $\mathbb{Q}[x]$. I used reduction mod p and for every prime p I found a factorizarion ...
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Prove that $6x^5-55x^3 + 50x^2+15$ is irreducible over $\mathbb{Q}[i]$ [duplicate]
We need to prove that $6x^5-55x^3 + 50x^2+15$ is irreducible over $\mathbb{Q}[i]$. We let $f(x) = 6x^5-55x^3 + 50x^2+15$. We use Eisenstein's criterion. We know that $(5)$ is a prime ideal in $\mathbb{...
- 352
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Show $x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Q}(\sqrt[5]{2})$
I have proved that $f(x)=x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Z}_2$, so it's irreducible in $\mathbb{Z}$, hence in $\mathbb{Q}$.
I know $f(x)=x^5+x^4+x^2+x+1$ is irreducible in $\mathbb{Q}(\...
- 311
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Proving that if $\ g(x$) is irreducible then $\ g(f(x))$ is also irreducible
Let $g(x)=ax^2+bx+1\in\mathbb{Z}[x]$ and
$f(x)=(x-a_1)(x-a_2)...(x-a_n), n\geq 7$ ($a_i\in\mathbb{Z}$ are
pairwise distinct). Prove that in $\mathbb{Q}[x]$, if $g(x)$ is
irreducible then $g(f(x))$ is ...
user1242284
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Why am I allowed to divide by polynomials even though multiplicative inverse do not exist?
Suppose I have the following equation:
$(x+1)(x-1)p(x) = (x-1)(x^2+x+1)q(x)$
According to the solutions I'm given, this is equivalent to
$(x+1)p(x) = (x^2+x+1)q(x)$
I get that, that one 'cancels out' $...
- 145
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Testing irreducibility of integer polynomials
I am currently maintaining a Julia software package CommutativeRings which implements also factorization of integer polynomials according to Bareiss and Zassenhaus. ...
- 31
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Find a cubic polynomial with given property.
Question. For a monic, irreducible cubic polynomial $p(x)\in \Bbb Z[x]$ with three real roots, prove or disprove that $p(x^3)$ must be irreducible over $\Bbb Z$ as well.
Here the notation $p(x^3)$ ...
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Most efficient way to show $X^3-8X+16$ is irreducible over $\mathbb{Q}(i)$.
I want to show that $X^3-8X+16$ is irreducible over $\mathbb{Q}(i)$. I'd like to know what the most efficient way to do this would be (i.e. shortest way to write something that works down in a way ...
- 3,940
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A faster way to check irreducibility of quartics in finite fields
A quadratic or a cubic can be shown to be irreducible in $\mathbb{F}_p$ by showing that none of $0, \dots,p-1$ are roots (or more generally all the elements of $\mathbb{F}_{p^n}$). This does not work ...
- 3,940
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Prove that the order of 2 modulo 2n+1 is odd
Let $n$ be a positive integer such that there exists a polynomial $f(x) \in \mathbb{F}_2[x]$ of degree n such that $f(x)\cdot x^n f(1/x) = 1+x+\cdots + x^{2n}\in \mathbb{F}_2[x]$. Prove that the order ...
- 1,225
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Prove that $x$ is irreducible in $\Bbb C[x,y]/(x^3+y^3-1)$
This is from the problem of proving $\Bbb C[x,y]/(x^3+y^3-1)$ is not a UFD. Here is my attempt:
Every polynomial in $\Bbb C[x,y]$ is of the form $$p_1(x)+p_2(x)y+p_3(x)y^2+r(x,y)(x^3+y^3-1)$$ for some ...
- 581
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How to prove a polynomial irreducibility result solved by R.Brauer.
We are familiar with the problem that
$$f(x)=(x-a_1)(x-a_2)\cdots(x-a_n)\pm1$$
and
$$f(x)=(x-a_1)^2(x-a_2)^2\cdots(x-a_n)^2+1$$
is irreducible with $a_i\in \mathbb{Z}$ and for the first problem, when $...
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