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Results tagged with polynomials
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user 6153
Questions in which polynomials (single or several variables) play a key role. It is typically important that this tag is combined with other tags; polynomials appear in very different contexts. Please, use at least one of the top-level tags, such as nt.number-theory, co.combinatorics, ac.commutative-algebra, in addition to it. Also, note the more specific tags for some special types of polynomials, e.g., orthogonal-polynomials, symmetric-polynomials.
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Is $x^p-x+1$ always irreducible in $\mathbb F_p[x]$?
In the notation of https://en.wikipedia.org/wiki/Berlekamp%27s_algorithm, the point is that the space of polynomials g congruent to their p-th power is not going to contain anything but constants, because …
- 12.6k
2
votes
Generators of cyclic group of finite fields
See for example https://en.wikipedia.org/wiki/Conway_polynomial_(finite_fields). There is plenty of theory and practice in the area, going back to Richard Parker.
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0
votes
Accepted
Probability of summing products of irreducible polynomials in a finite field to zero
Taking a view from old-fashioned algebra: your sum can be interpreted from a generating function of a product of three geometric series. That generating function has a partial fraction decomposition, …
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1
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minimal polynomials of trig functions of ($k \pi/p$) and divisibility of coefficients by p
I can't quite do this all in my head, but clearly enough you should should write 1 - ζ = π and do π-adic analysis in the cyclotomic field of p-th roots of unity, where ζ is a non-trivial p-th root of …
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5
votes
Which numbers appear as discriminants of cubics?
The discriminants of cubic polynomials are there explained in terms of discriminants of cubic fields, and asymptotics given for the latter. …
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1
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A question regarding polynomials whose roots satisfy certain algebraic relation
Such questions formed a substantial part of the classical "theory of equations", before Galois theory was formulated. For example, there is a book by Burnside on theory of equations, that is easy to f …
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1
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Roots of a polynomial in several variables
You need as many polynomials as variables to have a finite number. (There is much, much more to say.) …
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8
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What Are Some Naturally-Occurring High-Degree Polynomials?
Cyclotomic polynomials (http://en.wikipedia.org/wiki/Cyclotomic_polynomial), which when n is composite are rather more complicated than you might guess. … It is possible to reason about polynomials without knowing exactly what the solutions look like. …
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7
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What, if anything, makes homogeneous polynomials so great?
Well, if you are interested in counting solutions mod p, you could note that the "good" formulae are indeed related to the homogeneous approach/projective space. It is not just a question of restoring …
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