## Is F_P algebraically closed?

So first of all, there’s only one algebraically closed field of characteristic p that you should think about for the sake of intuition, the algebraic closure K of F_p. And that’s very easy, it’s just the limit of the finite fields of order p^n as n goes to infinity.

## Are complex numbers algebraically closed?

Equivalently (by definition), the theorem states that the field of complex numbers is algebraically closed. The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots.

**Are algebraic numbers closed under addition?**

The set of algebraic numbers is closed under addition, subtraction, multiplication and division. The set of algebraic integers is closed under addition, subtraction and multi- plication, but not division. 2. The root of a polynomial whose coefficients are algebraic numbers (resp., algebraic integers) is one also.

### Is C an algebraic closure of Q?

C is not an algebraic extension of Q, so by definition of algebraic closure it cannot be an algebraic closure of Q. The fact that this is a transcendental extension can be stated by proving, for instance, that e or π are not algebraic.

### Is an algebraic extension of an algebraic extension algebraic?

In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e. which contain transcendental elements, are called transcendental.

**How do you check if a number is an algebraic integer?**

Start with an initial guess G for O. Compute ΔG. For each prime p whose square devides ΔG, test all numbers of the form (I) to see which are algebraic integers. If any new integers arise, enlarge G to a new G′ by adding a new number.