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In this paper, I treat quadratic equation over associative $D$-algebra. In quaternion algebra $H$, the equation $x^2=a$ has either $2$ roots, or infinitely many roots. Since $ain R$, $a<0$, then the equation has infinitely many roots. Otherwise, the equation has roots $x_1$, $x_2$, $x_2=-x_1$. I considered different forms of the Vietes theorem and a possibility to apply the method of completing the square. In quaternion algebra, there exists quadratic equation which either has $1$ root, or has no roots.
From the symmetry between definitions of left and right divisors in associative $D$-algebra $A$, the possibility to define quotient as $Aotimes A$-number follows. In the paper, I considered division and division with remainder. I considered also definition of prime $A$-number.
In the book, I considered differential equations of order $1$ over Banach $D$Hyph algebra: differential equation solved with respect to the derivative; exact differential equation; linear homogeneous equation. I considered examples of differential eq
Module is effective representation of ring in Abelian group. Linear map of module over commutative ring is morphism of corresponding representation. This definition is the main subject of the book. To consider this definition from more general poin
Let $A$, $B$ be Banach $D$-algebras. The map $f:Arightarrow B$ is called differentiable on the set $Usubset A$, if at every point $xin U$ the increment of map $f$ can be represented as $$f(x+dx)-f(x) =frac{d f(x)}{d x}circ dx +o(dx)$$ where $$frac{d
Let $p$ be a large prime, and let $kll log p$. A new proof of the existence of any pattern of $k$ consecutive quadratic residues and quadratic nonresidues is introduced in this note. Further, an application to the least quadratic nonresidues $n_p$ modulo $p$ shows that $n_pll (log p)(log log p)$.