Algebraically closed field

2020 Mathematics Subject Classification: Primary: 12Exx Secondary: 12Fxx [MSN][ZBL]

A field $k$ is algebraically closed if any polynomial of non-zero degree over $k$ has at least one root in $k$. In fact, it follows that for an algebraically closed field $k$ each polynomial of degree $n$ over $k$ has exactly $n$ roots in $k$, i.e. each irreducible polynomial from the ring of polynomials $k[x]$ is of degree one. A field $k$ is algebraically closed if and only if it has no proper algebraic extension (cf. Extension of a field). For any field $k$, there exists a unique (up to isomorphism) algebraic extension of $k$ that is algebraically closed; it is called the algebraic closure of $k$ and is usually denoted by $\bar k$. Any algebraically closed field containing $k$ contains a subfield isomorphic to $k$.

The field of complex numbers is the algebraic closure of the field of real numbers. This is the fundamental theorem of algebra (cf. Algebra, fundamental theorem of).

References

[La]	S. Lang, "Algebra", Addison-Wesley (1974) MR0783636 Zbl 0712.00001
[ZaSa]	O. Zariski, P. Samuel, "Commutative algebra", 1, Springer (1975) MR0384768 Zbl 0313.13001