Question

Show that if \(E\) is an algebraic extension of \(K,\) and \(K\) is an algebraic extension of \(F,\) then \(E\) is an algebraic extension of \(F\).

Short answer

Answer: Yes, E is an algebraic extension of F.

Step-by-step solution

Recall the definition of an algebraic extension

An extension field K of a field F is called an algebraic extension if every element α in K is algebraic over F. This means that there exists a non-zero polynomial f(x) with coefficients from F such that f(α) = 0.

Provide an arbitrary element in E

To demonstrate that E is an algebraic extension of F, we need to show that for every element α in E, there exists a non-zero polynomial g(x) with coefficients from F, such that g(α) = 0. Let α be an arbitrary element in E.

Show that α is algebraic over K

Since E is an algebraic extension of K, there exists a non-zero polynomial f(x) ∈ K[x] such that f(α) = 0. Let's denote the degree of f(x) as n, and its coefficients as \(a_0, a_1, ..., a_n\), where \(a_i\) ∈ K for all i.

Show that K is algebraic over F

Since K is an algebraic extension of F, every element in K, including the coefficients of f(x), is algebraic over F. For each coefficient \(a_i\) ∈ K, there exists a non-zero polynomial \(h_i(x)\) ∈ F[x] with a degree of \(m_i\) such that \(h_i(a_i) = 0\).

Create a polynomial with coefficients from F

We can construct a new polynomial g(x) ∈ F[x] with coefficients from F using the coefficients of f(x) and the polynomials \(h_i(x)\). To do this, we will take the product of all possible combinations of f(x) and \(h_i(x)\) and add them together. Let g(x) = \(∑_{i=0}^n (x-a_i)^{m_i} f_i(x)\), where \(f_i(x)\) are the elements of F[x] and m_i is the degree of \(h_i(x)\).

Show that g(α) = 0

Now, let's evaluate g(α): g(α) = \(∑_{i=0}^n (α-a_i)^{m_i}f_i(α)\). Since \(f(α)=0\), this implies that g(α) = 0.

Conclude that E is an algebraic extension of F

Since α was an arbitrary element of E, we have shown that there exists a non-zero polynomial g(x) with coefficients from F such that g(α) = 0. Therefore, E is an algebraic extension of F.

Key concepts

Field Theory

Understanding field theory is essential to grasping the fundamentals of algebraic extensions. A field is a mathematical structure in which we can perform addition, subtraction, multiplication, and division (except by zero), and it adheres to properties like commutativity, associativity, and distributivity. Classical examples of fields include the set of rational numbers, real numbers, and complex numbers.

In field theory, we often explore the relationships between fields, especially how one field can be built on top of another. This leads to concepts like field extensions, where one field (called the extension field) contains another field (called the base field) and adds more elements to it, creating a larger field structure. For example, the complex numbers form an extension field to the real numbers. The study of how these fields relate and interact is crucial for higher mathematical concepts such as algebraic geometry, number theory, and Galois theory.

Polynomials

A polynomial is an expression consisting of variables and coefficients, using the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial is \( f(x) = 3x^2 + 2x - 5 \). Polynomials are critical in understanding algebraic extensions because they provide a means to define whether elements in an extension field are algebraic over the base field.

An element \( \alpha \) in an extension field is said to be algebraic over the base field if there exists a polynomial with coefficients in the base field that has \( \alpha \) as a root. This concept is utilized in the step-by-step solution to show how each element of an extension field can be related back to the base field through a specific polynomial. Understanding polynomials also involves knowing about their degree (the highest exponent of the variable), roots (solutions to \( f(x) = 0 \) ), and coefficients (the numbers in front of the variables).

Extension Field

An extension field is a broader field that contains a base field as a subset. The idea is that the extension field builds upon the base field by adding new elements, while still preserving the field properties for the entire set. In the context of the exercise, we are looking at algebraic extension fields, which mean that every new element in the extension field is algebraic over the base field.

The solution demonstrates that if \( E \) extends \( K \) and \( K \) extends \( F \), then every element in \( E \) is algebraic over \( F \). In other words, \( E \) is also an algebraic extension of \( F \). The noteworthy part of this process is constructing a polynomial with coefficients in \( F \) that has \( \alpha \) in \( E \) as a root, thereby proving \( E \) extends \( F \) algebraically. This is significant for advancing studies in algebra, where such properties of fields are applied to solve problems in space geometry, encryption algorithms, and error correction in data transmission.