as shown by the previous example, when the coefficients are explicit integers, the associated monic polynomial is generally more complicated. Therefore, primitive polynomials are often used instead of monic polynomials when dealing with integer coefficients. Integral elements Monic polynomial equations are at the basis of the theory of algebraic integers, and, more generally of integral elements. Let be a subring of a field ; this implies that is an integral domain. An element of is integral over if it is a root of a monic polynomial with coefficients in . 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 numbers that are also algebraic integers. This results from the rational root theorem, which asserts that, if the rational number is a root of a polynomial with integer coefficients, then is a divisor of the leading coefficient; so, if the polynomial is monic, then and the number is an integer. Conversely, an integer is a root of the monic polynomial It can be proved that, if two elements of a field are integral over a subring of , then the sum and the product of these elements are also integral over . It follows that the elements of that are integral over form a ring, called the integral closure of in . 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. Multivariate polynomials Ordinarily, the term monic is not employed for polynomials of several variables. However, a polynomial in several variables may be regarded as a polynomial in one variable with coefficients being polynomials in the other variables. Being monic depends thus on the choice of one "main" variable. For example, the polynomial is monic, if considered as a polynomial in with coefficients that are polynomials in : but it is not monic when considered as a polynomial in with coefficients polynomial in : In the context of Gröbner bases, a monomial order is generally fixed. In this case, a polynomial may be