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The Eisenstein criterion is well-known to all students of algebra, and it gives a very simple proof of the irreducibility of the cyclotomic polynomial Θ p (x)=x p-1 +x p-2 +⋯+1 for p prime. The irreducibility was (of course) known to Gauss but via a much more complicated proof. Simple ideas almost always arrive through tortuous detours, and the Eis...
Citations
... The irreducibility criteria due to Schönemann [11], Eisenstein [2], Perron [6], and Dumas [3] are among the most popular classical results concerning irreducibility of polynomials having integer coefficients [1]. The irreducibility criteria due to Schönemann and Eisenstein are easy consequences of the much more general factorization result of Dumas based on Newton-polygon approach. ...
... Testing polynomials for irreducibility over a given domain is an arduous task. Of particular interest are the polynomials having integer coefficients for which some well-known classical irreducibility criteria due to Sch€ onemann, Eisenstein, and Dumas exist (see [2,3,7] and for an insightful historical account of Sch€ onemann and Eisenstein criteria, see [1]). Recently, the elegant criteria established in [4,6] turn out to be extremely significant keeping in view their intimate connection with prime numbers. ...
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside a closed disk centered at the origin in the complex plane and deduce the irreducibility over the ring of integers.
... Testing polynomials for irreducibility over a given domain is an arduous task. Of particular interest are the polynomials having integer coefficients for which some well-known classical irreducibility criteria due to Schönemann, Eisenstein, and Dumas exist (see [1,2,4] and for an insightful historical account of Schönemann and Eisenstein criteria, see [3]). Recently, the elegant criteria established in [5,6] turn out to be extremely significant keeping in view their intimate connection with prime numbers. ...
In this article, we propose a few sufficient conditions on polynomials having integer coefficients all of whose zeros lie outside the closed unit disc centered at the origin in the complex plane and deduce the irreducibility over the ring of integers.
... The following result is fundamental (see, e.g., [Pra10, Theorem 2.1.3]) and follows from a result due to Schönemann (see, e.g., [Cox11]). The following result is well-known in the literature on FLT (see [Rib99] and references therein). ...
Given integers $\ell > m >0$, we define monic polynomials $X_n$, $Y_n$, and $Z_n$ with the property that $\mu$ is a root of $X_n$ if and only if the triple $(\mu,\mu+m,\mu+\ell)$ satisfies $x^n + y^n = z^n$. It is shown, in a precise asymptotic sense, that for a vast majority of cases, these polynomials are irreducible via Eisenstein's criterion.
... A more powerful, and yet easy to use, tool is Eisenstein's criterion: assuming that ( ) has integer coefficients, if for some prime number , the coefficients satisfy | for all ≠ , as well as ∤ and 2 ∤ 0 , then ( ) is irreducible. In fact, a statement equivalent to Eisenstein's Criterion was first proved by Schönemann [43] in a 1846 paper that Eisenstein even cites in his own paper [11] in 1850, hence the criterion was often called the Schönemann-Eisenstein theorem in literature from the early twentieth century, see [5] for a discussion. ...
... However, Hopper goes on to prove bounds on how far apart More precisely, Hopper establishes an upper bound, as in Theorem 1, that depends on the sharpness of the bends in ∞ ( ) ∘ ∞ (ℎ), defined as the (exponential of the) ratio of slopes of consecutive sides. Near very sharp bends, the two polygons are very close; the careful analysis [19,Chapter III,§5] of bends with small sharpness gives the upper bound. She remarks that the "result can however probably be considerably improved upon" due to certain estimates employed in the proof [19, p. 38]. ...
... The well-known fact that Eisenstein polynomials are irreducible is often encountered in an undergraduate algebra course. See [1] for a fascinating history of this result, which was proved independently by Schönemann and Eisenstein. Dobbs and Johnson (see [2]) posed some probabilistic questions concerning Eisenstein polynomials. ...
Previously Heyman and Shparlinski gave an asymptotic formula with error term for the number of Eisenstein polynomials of fixed degree and bounded height. Let $\psi(f)$ denote the number of primes for which a polynomial $f$ is Eisenstein. We give expressions for the mean and variance of the function $\psi$ for each fixed degree, where the polynomials are ordered according to their height.
... Over the years, these criteria have witnessed many variations and generalizations using prime ideals, valuations and Newton polygons (see [12], [9, Corollary 1.2], [2, Proposition 3.1], [4], [3], [11], [5]). In 2013, Weintraub [14] gave the following simple but interesting generalization of Eisenstein Irreducibility Criterion in an attempt to correct a false claim made by Eisenstein himself. ...
In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields.
... (b) 6 ∈ NCL(3) but 6 / ∈ NCL (2). Furthermore, if a 1 , . . . ...
... (iii) If n ≥ 8 is an even integer, then n ∈ NCL(2). (iv) If n ≥ 8 is an odd integer, then n ∈ NCL (2). ...
... Note that if the regular n-gon is non-constructible, then n ∈ NCL(1). However, if the regular n-gon is constructible, and infinitely many n ≥ 8 are such, then n / ∈ NCL(1) and, for these n, parts (iii) and (iv) above cannot be strengthened by changing NCL (2) to NCL (1). The elementary method we use to prove Part (iii) of Theorem 1.1 easily leads us to the following statement on cyclic n-gons of even order ; see Figure 1 for an illustration. ...
We study convex cyclic polygons, that is, inscribed n-gons. Starting from P. Schreiber's idea, published in 1993, we prove that these polygons are not constructible from their side lengths with straightedge and compass, provided n is at least five. They are non-constructible even in the particular case where they only have two different integer side lengths, provided that n 6= 6. To achieve this goal, we develop two tools of separate interest. First, we prove a limit theorem stating that, under reasonable conditions, geometric constructibility is preserved under taking limits. To do so, we tailor a particular case of Puiseux's classical theorem on some generalized power series, called Puiseux series, over algebraically closed fields to an analogous theorem on these series over real square root closed fields. Second, based on Hilbert's irreducibility theorem, we give a rational parameter theorem that, under reasonable conditions again, turns a non-constructibility result with a transcendental parameter into a non-constructibility result with a rational parameter. For n even and at least six, we give an elementary proof for the non-constructibility of the cyclic n-gon from its side lengths and, also, from the distances of its sides from the center of the circumscribed circle. The fact that the cyclic n-gon is constructible from these distances for n = 4 but non-constructible for n = 3 exemplifies that some conditions of the limit theorem cannot be omitted.
... Our approach is based on the following well-known statement from classical algebra. Its Part (C) is the Eisenstein-Schönemann criterion, see Cox [1] for our terminology. The degree of a polynomial f (x) in the variable x is denoted by deg x (f ). ...
... Observe that f (1) p (x) is a polynomial since p − j and n − p − j are even for j odd. ...
... p (x)/x) = p − 1 is not a power of 2. Since a, b ∈ Z, we can apply Proposition 2.1(A) (with s = 0 since no parameter is given) to f (1) p (x)/x to conclude that P n is not constructible. Alternatively, we can apply Proposition 2.1(A) and (B). ...
We deal with convex cyclic polygons with even order, that is, with inscribed
n-gons where n is even. We prove that these polygons are in general not
constructible with compass and ruler, provided n is at least six and even. We
conjecture that the statement also holds for odd orders. Some related questions
are also discussed.
... For an interesting review of the history of these results, and of some of the techniques used in their proof, as well as in the proof of some other important achievements of the 19th century number theory, we refer the reader to [7]. Over the years many authors contributed to the development of the techniques used in the study of the irreducibility of polynomials, some of them generalizing in various ways these irreducibility criteria. ...
The famous irreducibility criteria of Sch\"onemann-Eisenstein and Dumas rely
on information on the divisibility of the coefficients of a polynomial by a
single prime number. In this paper we provide several irreducibility criteria
of Sch\"onemann-Eisenstein-Dumas-type for polynomials with integer
coefficients, criteria that are given by some divisibility conditions for their
coefficients with respect to arbitrarily many prime numbers. A special
attention will be paid to those irreducibility criteria that require
information on the divisibility of the coefficients by two distinct prime
numbers.
