Topic: Mathematics \ Abstract Algebra \ Field Theory

Description:

Field Theory is a branch of Abstract Algebra dedicated to the study of fields. A field is a fundamental algebraic structure in which one can perform addition, subtraction, multiplication, and division (excluding division by zero), all of which conform to a set of arithmetic rules. Formally, a field \( F \) is a set equipped with two operations, typically referred to as addition (+) and multiplication (×), satisfying the following properties:

1. Closure: For all \( a, b \in F \),
   • \( a + b \in F \)
   • \( a \times b \in F \)
2. Associativity: For all \( a, b, c \in F \),
   • \( (a + b) + c = a + (b + c) \)
   • \( (a \times b) \times c = a \times (b \times c) \)
3. Commutativity: For all \( a, b \in F \),
   • \( a + b = b + a \)
   • \( a \times b = b \times a \)
4. Identity elements:
   • There exists an element \( 0 \in F \) such that \( a + 0 = a \) for all \( a \in F \).
   • There exists an element \( 1 \in F \) (with \( 1 \ne 0 \)) such that \( a \times 1 = a \) for all \( a \in F \).
5. Inverse elements:
   • For every \( a \in F \), there exists an element \( -a \in F \) such that \( a + (-a) = 0 \).
   • For every \( a \in F \) with \( a \ne 0 \), there exists an element \( a^{-1} \in F \) such that \( a \times a^{-1} = 1 \).
6. Distributivity: For all \( a, b, c \in F \),
   • \( a \times (b + c) = (a \times b) + (a \times c) \).

Field Theory often investigates special types of fields, such as finite fields (also called Galois fields), which have a finite number of elements. A classical example is the field of complex numbers, \( \mathbb{C} \), but other examples include the field of rational numbers, \( \mathbb{Q} \), and finite fields like \( \mathbb{F}_p \) (where \( p \) is a prime number).

One key problem in Field Theory is the characterization and classification of field extensions, which involve fields that are created by adding new elements to an existing field \( F \). A field extension \( E \) of \( F \) is written as \( E/F \). When the dimension of \( E \) as a vector space over \( F \) (denoted \([E:F]\)) is finite, \( E \) is called a finite extension of \( F \).

Key Concepts:

1. Field Extensions: If \( E \) is a field containing \( F \), then \( E \) is called an extension of \( F \), and is denoted by \( E/F \).
2. Algebraic and Transcendental Extensions: If an element \( \alpha \in E \) is a root of some non-zero polynomial with coefficients in \( F \), \( \alpha \) is called algebraic over \( F \). Otherwise, \( \alpha \) is transcendental over \( F \).
3. Galois Theory: A powerful branch of Field Theory that studies field extensions through group theory. Named after Évariste Galois, it provides profound connections between fields and groups.
4. Finite Fields: Fields with a finite number of elements, typically written as \( \mathbb{F}_q \), where \( q \) is a prime power \( p^n \).

Field Theory furnishes important tools and concepts used across various domains of mathematics, including Number Theory, Algebraic Geometry, and Cryptography. Its abstract nature and the richness of its structure make it a central and highly engaging area of study within Mathematics.