Editing Monic polynomial (section)

==Integral elements==
Monic polynomial equations are at the basis of the theory of [[algebraic integer]]s, and, more generally of [[integral element]]s.

Let {{mvar|R}} be a subring of a [[field (mathematics)|field]] {{mvar|F}}; this implies that {{mvar|R}} is an [[integral domain]]. An element {{mvar|a}} of {{mvar|F}} is ''integral'' over {{mvar|R}} if it is a [[polynomial root|root]] of a monic polynomial with coefficients in {{mvar|R}}.

A [[complex number]] that is integral over the integers is called an [[algebraic integer]]. This terminology is motivated by the fact that the integers are exactly the [[rational number]]s that are also algebraic integers. This results from the [[rational root theorem]], which asserts that, if the rational number <math display =inline>\frac pq</math> is a root of a polynomial with integer coefficients, then {{mvar|q}} is a divisor of the leading coefficient; so, if the polynomial is monic, then <math>q=\pm 1,</math> and the number is an integer. Conversely, an integer {{mvar|p}} is a root of the monic polynomial <math>x-a.</math>

It can be proved that, if two elements of a field {{mvar|F}} are integral over a subring   {{mvar|R}} of {{mvar|F}}, then the sum and the product of these elements are also integral over {{mvar|R}}. It follows that the elements of {{mvar|F}} that are integral over {{mvar|R}} form a ring, called the [[integral closure]] of {{mvar|R}} in {{mvar|K}}. An integral domain that equals its integral closure in its [[field of fractions]] is called an [[integrally closed domain]].

These concepts are fundamental in [[algebraic number theory]]. For example, many of the numerous wrong proofs of the [[Fermat's Last Theorem]] that have been written during more than three centuries were wrong because the authors supposed wrongly that the algebraic integers in an [[algebraic number field]] have [[unique factorization]].