All Questions
Tagged with polynomials irreducible-polynomials
1,520
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Find the value of $(1+\alpha)(1+\beta)(1+r)(1+s)$
If $\alpha,\beta,r,s$ are the roots of $x^{4}-x^{3}+x^{2}+x+3=0$, find the value of $(1+\alpha)(1+\beta)(1+r)(1+s)$.
This question appeared on one of the sample papers which I came across.
My first ...
3
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1
answer
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Characterization of irreducible polynomials over finite fields - alternative proof?
By accident I have found the following characterization of irreducible polynomials over $\mathbb{F}_p$.
Lemma. Let $g \in \mathbb{F}_p[T]$ be a monic polynomial of degree $m \geq 1$. Then, $g$ is ...
- 165k
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If a monic polynomial is irreducible in $\mathbb Z_{p^k}[X]$, then it is irreducible in $\mathbb Z_p[X]$
Let $p$ be a prime and $k$ is a positive integer. There is a natural projection of rings $Z_{p^k}[X] \longrightarrow Z_{p}[X]$. Suppose a monic polynomial $f(X) \in Z_{p^k}[X]$. $f(X)$ is called basic ...
- 1,201
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Is $x^6 + bx^3 + b^2$ irreducible?
Let $b\in \mathbb{Q}^*$ be rational number. We factorise $x^9-b^3\in \mathbb{Q}[x]$ and obtain $$x^9-b^3=(x^3-b)(x^6+bx^3+b^2).$$
Is the polynomial $x^6+bx^3+b^2$ irreducible?
If $b=1$ we get a ...
- 3,979
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Generalization of Eisenstein's Criterion [duplicate]
Let $f(X)=a_{2n+1}X^{2n+1}+\ldots+a_0\in \mathbb{Z}[X]$ with
$$\begin{align*}
a_{2n+1}&\not \equiv 0 \pmod p\\
a_{2n},\ldots,a_{n+1} &\equiv 0 \pmod p\\
a_n,\ldots,a_0&\equiv 0 \pmod{p^2} ...
- 2,234
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Reduce Polynomial Over Real Numbers
I was given the question $x^8 + 16$ and told to reduce it as much as able over the real numbers.
Here is what I tried
$x^8 + 16$
$(x^4+4)^2-8x^4$
$(x^4+4-2^{3/2}x^2)(x^4+4+2^{3/2}x^2)$
I can not ...
- 203
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$p$ a prime satisfying $p \equiv 3 \mod 4 $. Then, the quotient field $ F_p [x] / (x^2 + 1)$ contains $\bar{x}$ that is a square root of -1
I know that $x^2 + 1$ is irreducible in $F_p[x]$ if and only if $-1$ is not a square in $F_p$. Otherwise, $x^2 + 1$ could be factored out.
$-1$ not being a quadratic residue in $F_p$ is equivalent to ...
- 155
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On Unique factorization of polynomials
I'm studying Lang's Linear Algebra and stumbled upon a lemma prior to the unique factorization of polynomials that says the following "Let p be irreducible in K[t]. Let f, g be non-zero ...
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Generalization or criteria for a proposition for checking irreducibility of polynomials with summand of only two degrees
Is the follow proposition
Prop Let $f \in k\left[x_1, x_2, \ldots, x_n\right]$ be a polynomial with the form $l+h$, where $l$ is an irreducible non constant homogeneous polynomial and $h$ is a ...
- 1,268
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Question about the solution of the polynomial $(x−1)(x−2)⋯(x−n)−1$ is irreducible in $\mathbb{Z}\left [ x \right ]$ for all $n≥1$
The solution of the polynomial $(x−1)(x−2)⋯(x−n)−1$ is irreducible in $\mathbb{Z}\left [ x \right ]$ for all $n≥1$ is in here.
I think it's no problem that do the same thing on $\mathbb{Q\left [ x \...
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Reducibility of constrained polynomial
Let $f \in \mathbb{Z}[x, y]$ be a polynomial. Suppose that the list of terms of $f$ do not involve the $y$ variable except for a single $y^2$ term with some arbitrary coefficient. When is $f$ ...
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Is there a special name for linear irreducible polynomials (over the complex numbers)?
According to the fundamental theorem of algebra every polynomial over the complex numbers can be factorized into the following form:
$$
c (x - r_1) (x - r_2) (x - r_3) \dots
$$
where $r_i$ are the ...
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Does $\sqrt a + \sqrt b$ have a four way conjugate?
Let $a, b$ be rational numbers that are not perfect squares. Consider the set $S = \{\sqrt a + \sqrt b, \sqrt a - \sqrt b, - \sqrt a + \sqrt b, -\sqrt a - \sqrt b\}$.
If $p$ is a polynomial with ...
- 4,450
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$x^6 + 69x^5 − 511x + 363$ is irreducible over $\mathbb Z$?
As mentioned, I am trying to show that $x^6 + 69x^5 − 511x + 363$ is irreducible over $\mathbb Z$. To see that it has no roots and no cubic factors, I send the polynomial to $\mathbb F_7$ and $\mathbb ...
- 1,042
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Is $13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i$ irreducible in $(\mathbb{Q}[i])[x]$?
Is $$13x^5 + (3 − i)x^3 + (8 − i)(x^2 − x) + 1 − 2i$$ irreducible in $(\mathbb{Q}[i])[x]$?
I've tried using Eisenstein’s irreducibility criterion to prove that it is, but I don't think it applies ...
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