Survey on irreducibility of trinomials
Let $a,b,c$ be non-zero integers and $f(x)=ax^n+bx^m+c$ be a trinomial of degree $n$. We surveyed the irreducibility criteria of $f(x)$ over rational numbers.
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arXiv subjects
Biswajit Koley
Publications and source records attributed to Biswajit Koley.
Irreducibility of $x^n-a$
A. Capelli gave a necessary and sufficient condition for the reducibility of $x^n-a$ over $\Q$. In this article, we are providing an alternate elementary proof for the same.
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An irreducible class of polynomials over integers
In this article, we consider polynomials of the form $f(x)=a_0+a_{n_1}x^{n_1}+a_{n_2}x^{n_2}+\dots+a_{n_r}x^{n_r}\in \mathbb{Z}[x],$ where $|a_0|\ge |a_{n_1}|+\dots+|a_{n_r}|,$ $|a_0|$ is a prime power and $|a_0|\nmid |a_{n_1}a_{n_r}|$. We will show that under the strict inequality these polynomials are irreducible for certain values of $n_1$. In the case of equality, apart from its cyclotomic factors, they have exactly one irreducible non-reciprocal factor.
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Irreducibility criterion for certain trinomials
In this article we study the irreducibility of polynomials of the form $x^n+ε_1 x^m+p^kε_2$, $p$ being a prime number. We will show that they are irreducible for $m=1$. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the form $x^n+ε_1x^m+ε_2$, where $ε_i\in \{\, -1,+1\,\}$. This corrects few of the existing results of W. Ljuggren's on $x^n+ε_1x^m+ε_2$.
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An irreducibility criterion for polynomials over integers
In this article, we consider the polynomials of the form $f(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n\in \mathbb{Z}[x],$ where $|a_0|=|a_1|+\dots+|a_n|$ and $|a_0|$ is a prime. We show that these polynomials have a cyclotomic factor whenever reducible. As a consequence, we give a simple procedure for checking the irreducibility of trinomials of this form and separability criterion for certain quadrinomials.
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