Rings in which every monic polynomial has a root, are called absolutely integrally closed.

Such rings are not necessarily a field: For example the ring of all algebraic integers (that is the ring all roots of monic polynomials with rational integral coefficients) is absolutely integrally closed, but no field.

For more Information on absolutely integrally closed rings see https://stacks.math.columbia.edu/tag/0DCK. See also Section 4.7 of Swanson's and Huneke's book https://www.math.purdue.edu/~iswanso/book/SwansonHuneke.pdf for results on the absolute integral closure of a reduced ring with finitely many minimal primes.

Rings in which every monic polynomial has a root, are called absolutely integrally closed.

Such rings are not necessarily a field: For example the ring of all algebraic integers (that is the ring all roots of monic polynomials with rational integral coefficients) is absolutely integrally closed, but no field.

For more Information on absolutely integrally closed rings see https://stacks.math.columbia.edu/tag/0DCK. See also Section 4.7 of the book "Swanson, Huneke: Integral Closure of Ideals, Rings and Modules" (LMS, Lecture Notes Series 336), online: https://www.math.purdue.edu/~iswanso/book/SwansonHuneke.pdf for results on the absolute integral closure of a reduced ring with finitely many minimal primes.

Rings in which every monic polynomial has a root, are called absolutely integrally closed.

Such rings are not necessarily a field: For example the ring of all algebraic integers (that is the ring all roots of monic polynomials with rational integral coefficients) is absolutely integrally closed, but no field.

For more Information on absolutely integrally closed rings see https://stacks.math.columbia.edu/tag/0DCK. See also Section 4.7 of Swanson's and Huneke's book http://people.reed.edu/~iswanson/book/SwansonHuneke.pdf for results on the absolute integral closure of a reduced ring with finitely many minimal primes.

Rings in which every monic polynomial has a root, are called absolutely integrally closed.

Such rings are not necessarily a field: For example the ring of all algebraic integers (that is the ring all roots of monic polynomials with rational integral coefficients) is absolutely integrally closed, but no field.

For more Information on absolutely integrally closed rings see https://stacks.math.columbia.edu/tag/0DCK. See also Section 4.7 of Swanson's and Huneke's book https://www.math.purdue.edu/~iswanso/book/SwansonHuneke.pdf for results on the absolute integral closure of a reduced ring with finitely many minimal primes.