examples of integrally closed extensions
Example. is not integrally closed, for is integral over since , but .
Example. is not integrally closed. Note that , but that
and so is integral over since it satisfies the polynomial .
The kernel of this map is , and its image is . Hence
and the field of fractions of the latter ring is obviously . Now, is integral over ( is its polynomial), but is not in . corresponds to in the original ring , which is thus not integrally closed (the minimal polynomial of is since ). The failure of integral closure in this coordinate ring is due to a codimension 1 singularity of at .
Example. is integrally closed. For again, parameterize by
The kernel of this map is and its image is . Claim is integrally closed. We prove this by showing that the integral closure of in is . Choose such that is integral over . Then is also integral over , so their sum is. Hence is integral over . But is a UFD, hence integrally closed, so and thus . Similarly, is integral over , hence . Clearly, then, can have no denominator, so . Hence .
|Title||examples of integrally closed extensions|
|Date of creation||2013-03-22 17:01:32|
|Last modified on||2013-03-22 17:01:32|
|Last modified by||rm50 (10146)|