Irreducibility of Polynomials

Dorwart, Harold L.

doi:10.2307/2301357

Cited by 11 publications

(14 citation statements)

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“…Brauer [5] and Dorwart [10] have investigated the situation more closely, see Dorwart [11] for more information. Weisner [46] studied for general nonzero integer c the polynomials of degree n which assume the value c for n distinct integral values of x.…”

Section: Remark 42mentioning

confidence: 99%

Irreducibility criteria of Schur-type and Pólya-type

2010

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Section: Remark 42mentioning

confidence: 99%

Irreducibility criteria of Schur-type and Pólya-type

2010

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“…We write f(x) = x 7 + 20A: 6 We prove the irreducibility of f(x) andf(-x) by using the method of G. Dumas [3] (cf. Dorwart [2]) based on Newton polygon. The Newton polygon corresponding to/(x) for the prime 2 is shown on Figure 1.…”

Section: Proofmentioning

confidence: 99%

On the number of terms in the irreducible factors of a polynomial over ℚ

Choudhry

Schinzel

1992

Glasgow Math. J.

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All polynomials considered in this paper belong to Q [x] and reducibility means reducibility over Q. It has been established by one of us [5] that every binomial in Q [x] has an irreducible factor which is either a binomial or a trinomial. He has further raised the question "Does there exist an absolute constant K such that every trinomial in Q [x] has a factor irreducible over Q which has at most K terms (i.e. K non-zero coefficients)?"A similar question could be asked for a quadrinomial, or, more generally, for a polynomial with m non-zero coefficients. This paper deals with the general problem, that could be formulated as follows:Given a positive integer m does there exist a number K such that every polynomial in Q [x] with m non-zero coefficients has a factor irreducible over Q with at most K non-zero coefficients (AT "terms")?If for a given m numbers K with the above property exist we denote by K{m) the least of them, otherwise we put K{m) = °°. We shall prove

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“…For a statement of the criterion, we turn to Dorwart's 1935 article "Irreducibility of polynomials" in the American Mathematical Monthly [9]. As you might expect, he begins with Eisenstein:…”

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confidence: 99%

“…Then Eisenstein proceeds to prove that W is irreducible using one of the standard proofs of the Eisenstein criterion. 9 In other words, Eisenstein's first proof of his criterion…”

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confidence: 99%

“…The word "Geheimrath," now spelled "Geheimrat," originated as the German equivalent of a "Privy councillor" in a governmental context and was an honorific for distinguished professors in German universities in the 19th century. 9 There are two standard proofs of the Eisenstein criterion. One proof (due to Eisenstein) works by studying which coefficients of the factors are divisible by the prime.…”

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confidence: 99%

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