# Real radical

In algebra, the **real radical** of an ideal *I* in a polynomial ring with real coefficients is the largest ideal containing *I* with the same vanishing locus.
It plays a similar role in real algebraic geometry that the radical of an ideal plays in algebraic geometry over an algebraically closed field.
More specifically, the Nullstellensatz says that when *I* is an ideal in a polynomial ring with coefficients coming from an algebraically closed field, the radical of *I* is the set of polynomials vanishing on the vanishing locus of *I*. In real algebraic geometry, the Nullstellensatz fails as the real numbers are not algebraically closed. However, one can recover a similar theorem, the *real Nullstellensatz*, by using the real radical in place of the (ordinary) radical.

## Definition

The **real radical** of an ideal *I* in a polynomial ring over the real numbers, denoted by , is defined as

The Positivstellensatz then implies that is the set of all polynomials that vanish on the real variety defined by the vanishing of .

## References

- Marshall, Murray Positive polynomials and sums of squares. Mathematical Surveys and Monographs, 146. American Mathematical Society, Providence, RI, 2008. xii+187 pp. ISBN 978-0-8218-4402-1; 0-8218-4402-4