Let (A,m) be an abstract complete intersection and let P a prime ideal of A. In [1] Avramov proved that AP is intersection. this article, we give elementary proof t...

Let $m$ and $n$ be positive integers, let $p$ a prime. $T(x)=\Phi_{p^m}(\Phi_{2^n}(x))$, where $\Phi_k(x)$ is the cyclotomic polynomial of index $k$. In this article, we prove that $T(x)$ irreducible over $\mathbf Q$ $\left\{1,\theta,\theta^2,\ldots,\theta^{2^{n-1}p^{m-1}(p-1)-1}\right\}$ basis for ring integers Q (\theta)$, $T(\theta)= 0$.

Let n be a positive natural number. Let us note that n− 1 is natural. Let n be a non trivial natural number. Observe that n− 1 is positive. Let x be an integer number and n be a natural number. Let us observe that xn is integer. Let us observe that 1n reduces to 1. Let n be an even natural number. Let us observe that (−1) reduces to 1. Let n be an odd natural number. One can verify that (−1) re...

Abstract For a ∈ (0, 1/2], r 1), let

Let $f(z)$ be an analytic function on the unit disk ${zinmathbb{C}, |z|leq 1}$, for each $q>0$, the $|f|_{q}$ is defined as followsbegin{align*}begin{split}&left|fright|_q:=left{frac{1}{2pi}int_0^{2pi}left|f(e^{itheta})right|^qdthetaright}^{1/q}, 0

For a character χ of finite group G, the co-degree is c (1)=[G:kerχ] χ(1). Let p be prime and let e positive integer. In this paper, we first show that if G p-solvable such e+1 ∤χ (1), for every irreducible then p-length not greater than e. Next, study groups satisfying condition 2 does divide co-degrees their characters.

Let G be a graph with p vertices and q edges and A = {0, 1, 2, . . . , [q/2]}. A vertex labeling f : V (G) → A induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. For a ∈ A, let vf (a) be the number of vertices v with f(v) = a. A graph G is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in A, |vf (a) − vf (b)| ≤ 1 and the indu...