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Field Extensions

  • Understand the definition and types of field extensions.
  • Learn about algebraic and transcendental extensions.
  • Explore examples of field extensions and their applications.

Definition of Field Extensions

A field extension is a larger field that contains a smaller field as a subfield. Formally, if \( F \) is a field and \( E \) is a field such that \( F \subseteq E \), then \( E \) is called a field extension of \( F \), and we write \( E/F \).

Types of Field Extensions

Field extensions can be classified into two main types:

  • Algebraic Extensions: A field extension \( E/F \) is algebraic if every element of \( E \) is a root of some non-zero polynomial with coefficients in \( F \). In other words, if \( \alpha \in E \) is algebraic over \( F \), there exists a polynomial \( f(x) \in F[x] \) such that \( f(\alpha) = 0 \).
  • Transcendental Extensions: A field extension \( E/F \) is transcendental if there exists at least one element in \( E \) that is not algebraic over \( F \). In other words, if \( \alpha \in E \) is transcendental over \( F \), there is no non-zero polynomial \( f(x) \in F[x] \) such that \( f(\alpha) = 0 \).

Examples of Field Extensions

  • Example 1: The field of complex numbers \( \mathbb{C} \) is an extension of the field of real numbers \( \mathbb{R} \). Every complex number can be expressed as \( a + bi \) where \( a, b \in \mathbb{R} \) and \( i \) is the imaginary unit.
  • Example 2: The field of rational numbers \( \mathbb{Q} \) is an extension of the field of integers modulo \( p \), \( \mathbb{Z}_p \), where \( p \) is a prime number.
  • Example 3: The field \( \mathbb{Q}(\sqrt{2}) \) is an algebraic extension of the field \( \mathbb{Q} \). Every element of \( \mathbb{Q}(\sqrt{2}) \) can be written as \( a + b\sqrt{2} \) where \( a, b \in \mathbb{Q} \).
  • Example 4: The field \( \mathbb{Q}(\pi) \) is a transcendental extension of the field \( \mathbb{Q} \) because \( \pi \) is a transcendental number (not a root of any non-zero polynomial with rational coefficients).

Exercises

  • Identify Field Extensions: Determine if \( \mathbb{Q}(\sqrt{3}) \) is an algebraic or transcendental extension of \( \mathbb{Q} \).
  • Field Operations: Perform addition and multiplication in the field \( \mathbb{Q}(\sqrt{5}) \) with given elements \( a + b\sqrt{5} \) and \( c + d\sqrt{5} \).
  • Polynomial Roots: Show that \( \sqrt{2} \) is algebraic over \( \mathbb{Q} \) by finding a polynomial with rational coefficients that has \( \sqrt{2} \) as a root.

Summary

In this lesson, we explored the concepts of field extensions, including their definitions, types, and examples. Understanding these fundamental ideas is crucial as they provide insights into the algebraic structure of fields and their applications in various areas of mathematics.

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