Polynomial Algebra and Field Extensions

Polynomial Algebra

• Polynomials over various fields (e.g., R[x] where R is a commutative ring) are fundamental in abstract algebra.
• Polynomial factorization varies across fields (real, complex, finite fields). For example, x² + 1 is irreducible over real numbers but factors as (x - i)(x + i) over complex numbers.
• Roots of polynomial equations lead to algebraic field extensions (e.g., solving x⁴ - 2 = 0 requires introducing √2).
• The minimal polynomial of an algebraic number α over field F is the monic polynomial of lowest degree with α as a root (e.g., x² - 2 is the minimal polynomial of √2 over ℚ).

Field Extensions

• A field extension E/F is an extension of a base field F to a larger field E (F ⊆ E).
• The degree of an extension [E:F] is the dimension of E as a vector space over F (e.g., [ℚ(√2):ℚ] = 2).
• An algebraic extension means every element of E is a root of a non-zero polynomial with coefficients in F (e.g., ℚ(√2) is an algebraic extension).
• Normal extensions contain all roots of their minimal polynomials; separable extensions have minimal polynomials with distinct roots.

Galois Theory

• Galois theory explores the symmetries of polynomial equation roots, connecting algebra and geometry.
• The Galois group Gal(E/F) comprises automorphisms of E fixing F; it reveals insights into polynomial solvability by radicals.
• The Fundamental Theorem of Galois Theory links intermediate fields of an extension with subgroups of its Galois group. For instance, the Galois group of ℚ(ζₙ)/ℚ (ζₙ is a primitive nth root of unity) is isomorphic to (ℤ/nℤ)*.

Linear Algebraic Equations

• Matrix decompositions (LU, QR, SVD) solve linear systems and compute matrix invariants. LU decomposition expresses A as LU (L is lower triangular, U is upper triangular).
• Eigenvalues (λ) and eigenvectors (v) satisfy Av = λv; they are crucial in physics and engineering. The characteristic polynomial, det(A - λI) = 0, yields eigenvalues.

Advanced Topics: Noetherian Rings and Modules

• A Noetherian ring R has every ideal finitely generated; this is essential in ideal theory and algebraic geometry (e.g., k[x₁, x₂, ..., xₙ] is Noetherian by Hilbert's basis theorem).
• Modules generalize vector spaces, using ring elements instead of field elements as scalars; they are applied in homological algebra and representation theory.