A polynomial f (x) ∈ F[x] is reducible over F if we can factor it as f (x) = g(x)h(x) for some g(x), h(x) ∈ F[x] of strictly lower degree. If f (x) is not reducible, we say it is irreducible over F.
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Apr 1, 2018 · There are several well-known criteria for a polynomial with integer coefficients to be irreducible over Z, e.g., Eisenstein's criterion. I'm ...
One says that the polynomial x2 − 2 is irreducible over the integers but not over the reals. Polynomial irreducibility can be considered for polynomials with ...
Jun 20, 2017 · If the polynomial is constant (and not ±1), then it is irreducible if and only if cf is prime.
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers ...
Sep 7, 2021 · A nonconstant polynomial f(x)∈F[x] is irreducible over a field F if f(x) cannot be expressed as a product of two polynomials g(x) and h(x) in F ...
A polynomial is called reducible if it can be factored into two or more non-constant polynomials. In this case, we want to prove that x 2 + 1 is reducible in Z ...
Nov 20, 2020 · A polynomial f e F[2] is called irreducible if the following conditions hold: (1) deg f > 1, i.e., f is not a constant polynomial, and (2) The ...
Mar 18, 2022 · I first checked for rational roots for this polynomial. The options are , both don't nullify the polynomial thus this polynomial doesn't have ...
We recall several different ways we have to prove that a given polynomial is irreducible. As always, k is a field. Theorem 0.1 (Gauss' Lemma).
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