Root_of_a_polynomial | TriviaQuestions.online

⭐ Core Definition: Root of a polynomial

In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function $f$ , is a member $x$ of the domain of $f$ such that $f(x)$ vanishes at $x$ ; that is, the function $f$ attains the value of 0 at $x$ , or equivalently, $x$ is a solution to the equation $f(x)=0$ . A "zero" of a function is thus an input value that produces an output of 0.

A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial $f$ of degree two, defined by $f(x)=x^{2}-5x+6=(x-2)(x-3)$ has the two roots (or zeros) that are 2 and 3. $f(2)=2^{2}-5\times 2+6=0{\text{ and }}f(3)=3^{2}-5\times 3+6=0.$

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👉 Root of a polynomial in the context of Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers $A$ is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.

The ring of integers of a number field $K$ , denoted by $O K$ , is the intersection of $K$ and $A$ : it can also be characterized as the maximal order of the field $K$ . Each algebraic integer belongs to the ring of integers of some number field. A number $α$ is an algebraic integer if and only if the ring $\mathbb {Z} [\alpha ]$ is finitely generated as an abelian group, which is to say, as a $\mathbb {Z}$ -module.

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⭐ Core Definition: Root of a polynomial
👉 Root of a polynomial in the context of Algebraic integer
Root of a polynomial in the context of Integral element
Root of a polynomial in the context of Ring of integers

Root of a polynomial in the context of Integral element

In commutative algebra, an element b of a commutative ring B is said to be integral over a subring A of B if b is a root of some monic polynomial over A.

If A, B are fields, then the notions of "integral over" and of an "integral extension" are precisely "algebraic over" and "algebraic extensions" in field theory (since the root of any polynomial is the root of a monic polynomial).

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Root of a polynomial in the context of Ring of integers

In mathematics, the ring of integers of an algebraic number field $K$ is the ring of all algebraic integers contained in $K$ . An algebraic integer is a root of a monic polynomial with integer coefficients: $x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}$ . This ring is often denoted by $O_{K}$ or ${\mathcal {O}}_{K}$ . Since any integer belongs to $K$ and is an integral element of $K$ , the ring $\mathbb {Z}$ is always a subring of $O_{K}$ .

The ring of integers $\mathbb {Z}$ is the simplest possible ring of integers. Namely, $\mathbb {Z} =O_{\mathbb {Q} }$ where $\mathbb {Q}$ is the field of rational numbers. And indeed, in algebraic number theory the elements of $\mathbb {Z}$ are often called the "rational integers" because of this.

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Root of a polynomial in the context of "Algebraic integer"

⭐ In the context of algebraic integers, a root of a polynomial is considered 'algebraic' when it satisfies a specific condition regarding the polynomial's structure. What must be true of the polynomial?

⭐ Core Definition: Root of a polynomial

👉 Root of a polynomial in the context of Algebraic integer

Root of a polynomial in the context of Integral element

Root of a polynomial in the context of Ring of integers