Start with the definition that you use: What is an algebraic closed field? What is an algebraic extension? How do they relate to F?Mr Davis 97 said:Homework Statement
Let E be an algebraic extension of F in which every polynomial in F[x]
can be factored into linear polynomials. Then E is algebraically closed.
Homework Equations
The Attempt at a Solution
This seems like a very easy problem, but I'm not sure how to write things down formally.
A field F is algebraically closed if every non-constant polynomial with coefficients in F has at least one root in F.
Alg ext is the set of all algebraic extensions of a given field F. In other words, it is the collection of all fields that can be obtained by adjoining algebraic elements to F.
Showing that Alg ext is closed when F is algebraically closed is important because it allows us to extend the properties of F to all of its algebraic extensions. This allows us to apply theorems and techniques from F to a larger class of fields, making our analysis more versatile and comprehensive.
We can prove that Alg ext is closed when F is algebraically closed by using the fundamental theorem of Galois theory. This theorem states that every algebraic extension of a field can be expressed as a splitting field of a separable polynomial over that field. Since F is algebraically closed, all polynomials over F are separable, and thus all algebraic extensions of F are splitting fields. Therefore, Alg ext is closed.
Some examples of algebraically closed fields include the complex numbers, the algebraic closure of the rational numbers, and the algebraic closure of finite fields. In general, any field can be made algebraically closed by adjoining all of its algebraic elements.