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This results from the rational root theorem, which asserts that, if the rational number is a root of a polynomial with integer coefficients, then q is a divisor of the leading coefficient; so, if the polynomial is monic, then =, and the number is an integer.
If () is a monic polynomial in one variable with coefficients in a unique factorization domain (or more generally a GCD domain), then a root of that is in the field of fractions of is in . [ note 5 ] If R = Z {\displaystyle R=\mathbb {Z} } , then it says a rational root of a monic polynomial over integers is an integer (cf. the rational root ...
For polynomials with real or complex coefficients, it is not possible to express a lower bound of the root separation in terms of the degree and the absolute values of the coefficients only, because a small change on a single coefficient transforms a polynomial with multiple roots into a square-free polynomial with a small root separation, and ...
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 element α has a minimal polynomial when α is algebraic over F, that is, when f(α) = 0 for some non-zero polynomial f(x) in F[x]. Then the minimal polynomial of α is defined as the monic polynomial of least degree among all polynomials in F[x] having α as a root.
In linear algebra, the minimal polynomial μ A of an n × n matrix A over a field F is the monic polynomial P over F of least degree such that P(A) = 0. Any other polynomial Q with Q(A) = 0 is a (polynomial) multiple of μ A. The following three statements are equivalent: λ is a root of μ A, λ is a root of the characteristic polynomial χ A ...
The minimal polynomial of an algebraic element x of L is irreducible, and is the unique monic irreducible polynomial of which x is a root. The minimal polynomial of x divides every polynomial which has x as a root (this is Abel's irreducibility theorem).
Quadratic irrational numbers, irrational solutions of a quadratic polynomial ax 2 + bx + c with integer coefficients a, b, and c, are algebraic numbers. If the quadratic polynomial is monic (a = 1), the roots are further qualified as quadratic integers. Gaussian integers, complex numbers a + bi for which both a and b are integers, are also ...