## One Citation

Rings and fields of constants of cyclic factorizable derivations

- MathematicsAnalytic and Algebraic Geometry 3
- 2019

We present a survey of the research on rings of polynomial constants and fields of rational constants of cyclic factorizable derivations in polynomial rings over fields of characteristic zero. 1.…

## References

SHOWING 1-10 OF 42 REFERENCES

R ings of constants for k-derivations in k(xi, xn)

- Mathematics
- 1988

1. Preliminaries. Let us recall at first ([1 ]) that if x n] is the ring of polynomials over a commutative ring k and f n e k [x 1 ,.. .;x 0 ] then there exists a unique k-derivation d of xn] such…

A FACTORISABLE DERIVATION OF POLYNOMIAL RINGS IN n VARIABLES

- Mathematics
- 2010

Let k[x1,….,xn] be the polynomial ring in n 3 variables over a field k of characteristic zero, and let be the factorisable derivation of k[x1,…., xn] defined by (xi) = xi(S - xi); for i = 1,….,…

On the Cyclotomic Polynomial Φpq (X)

- Mathematics
- 1996

The mth cyclotomic polynomial 4>m(X) is defined to be fl(X;), where; ranges over the primitive mth roots of unity in C:. It is well-known that 4>m(X) is an irreducible polynomial in @[X] with degree…

FINITE FIELDS

- Mathematics
- 2004

This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and Fp interchangeably for the…

On the coefficients of the Cyclotomic polynomial

- Mathematics
- 1946

For n < 105 all coefficients of F,(X) are zb 1 or 0. For n = 105, the coefficient 2 occurs for the first time. Denote by A, the greatest coefficient of F,,(x) (in absolute value). Schur proved that…

A note on the cyclotomic polynomial

- Mathematics
- 1964

The n -th roots of unity 1, ω, …, ω n - 1 , where ω = exp (2π i \ n ), are linearly dependent in the field Q of rationals since, for instance, their sum vanishes. We are here concerned with the…

On Vanishing Sums of Roots of Unity

- Mathematics
- 1995

An unsolved problem in number theory asked the following: For a given natural number m, what are the possible integers n for which there exist mth roots of unity α1,…,αn ∈ C such that α1 + ··· + αn =…

The Lowest-Degree Polynomial with Nonnegative Coefficients Divisible by the n-th Cyclotomic Polynomial

- MathematicsElectron. J. Comb.
- 2012

Determining the lowest-degree polynomial with nonnegative coefficients divisible by $\Phi_n(x)$ remains open in the general case, though it is conjecture the existence of values of $n$ for which this degree is, in fact, less than $(p-1)n/p).