2.2 Algebraic and transcendental elements

Algebraic and transcendental elements are key concepts in field extensions. Algebraic elements are roots of polynomials over the base field, while transcendental elements aren't. Understanding these helps classify field extensions and their properties.

Proving element types involves finding polynomials for algebraic elements or showing none exist for transcendental ones. This knowledge is crucial for constructing and analyzing different types of field extensions, which form the foundation of Galois theory.

Algebraic vs Transcendental Elements

Definition and Characteristics

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• An element α in a field extension L/K is algebraic over K when it is a root of some nonzero polynomial f(x) with coefficients in K
• An element α in a field extension L/K is transcendental over K when it is not algebraic over K, meaning there is no nonzero polynomial f(x) with coefficients in K such that f(α) = 0
• Every element in a field extension L/K is either algebraic or transcendental over K, but not both

Subfields and Extension Types

• The set of all elements in L that are algebraic over K forms a subfield of L called the algebraic closure of K in L
• If every element in a field extension L/K is algebraic over K, the extension is called an algebraic extension (Q(√2) over Q, where √2 is a root of x^2 - 2)
• If not every element in a field extension L/K is algebraic over K, the extension is called a transcendental extension (Q(π) over Q, where π is not a root of any nonzero polynomial with rational coefficients)

Proving Element Type

Proving Algebraic Elements

• To prove that an element α in a field extension L/K is algebraic over K, find a nonzero polynomial f(x) with coefficients in K such that f(α) = 0
  • The polynomial f(x) is called the minimal polynomial of α over K when it is monic, has the smallest degree among all polynomials with α as a root, and has coefficients in K
• The degree of the minimal polynomial of an algebraic element α over K is called the degree of α over K, denoted [K(α):K]

Proving Transcendental Elements

• To prove that an element α in a field extension L/K is transcendental over K, show that there is no nonzero polynomial f(x) with coefficients in K such that f(α) = 0
  • This can be done by assuming the existence of such a polynomial and deriving a contradiction, or by using the properties of transcendental elements
• If α is transcendental over K, then [K(α):K] is infinite

Constructing Extensions

Constructing Algebraic Extensions

• To construct an algebraic extension, adjoin a root of a polynomial with coefficients in the base field to the base field
• Examples of algebraic extensions include:
  • Q(i) over Q, where i is a root of the polynomial x^2 + 1
  • Finite fields F_p^n over F_p, where p is a prime and n is a positive integer

Constructing Transcendental Extensions

• To construct a transcendental extension, adjoin an element that is not a root of any nonzero polynomial with coefficients in the base field to the base field
• Examples of transcendental extensions include:
  • Q(e) over Q, where e is the base of the natural logarithm and is not a root of any nonzero polynomial with rational coefficients
  • The field of rational functions K(x) over a field K, where x is an indeterminate

Properties of Element Types

Properties of Algebraic Elements

• The set of algebraic elements in a field extension L/K forms a subfield of L
• If α is algebraic over K and β is algebraic over K(α), then β is algebraic over K (transitivity of algebraic extensions)
• If α and β are algebraic over K, then α + β, α - β, αβ, and α/β (if β ≠ 0) are also algebraic over K

Properties of Transcendental Elements and Extensions

• The set of transcendental elements in a field extension L/K does not form a subfield of L
• If α is transcendental over K, then K(α) is isomorphic to the field of rational functions K(x)
• The algebraic closure of a field K is the smallest algebraically closed field containing K, denoted K̄ (unique up to isomorphism and contains all the roots of polynomials with coefficients in K)
• Transcendental extensions are essential in the study of transcendental numbers, such as π and e, which have important applications in mathematics and other fields