Integral Element - Examples

Examples

• Integers are the only elements of Q that are integral over Z. In other words, Z is the integral closure of Z in Q.
• Gaussian integers, complex numbers of the form, are integral over Z. is then the integral closure of Z in .
• The integral closure of Z in consists of elements of form called Dirichlet integers; these are examples of quadratic integers.
• Let a positive integer. Then the integral closure of Z in the cyclotomic field is .
• The integral closure of Z in the field of complex numbers C is called the ring of algebraic integers.
• If is an algebraic closure of a field k, then is integral over
• Let a finite group G act on a ring A. Then A is integral over the set of elements fixed by G. see ring of invariants.
• The roots of unity and nilpotent elements in any ring are integral over Z.
• Let R be a ring and u a unit in a ring containing R. Then (i) is integral over if and only if (ii) is integral over R.
• The integral closure of in a finite extension of is of the form (cf. Puiseux series)
• The integral closure of the homogeneous coordinate ring of a normal projective variety X is the ring of sections