Field Theory

Field theory studies extensions of fields, algebraic elements, automorphisms, and Galois-theoretic structure.

Galois theory, with nearly half of the subject's articles, includes the Abel–Ruffini theorem that every complex root of every rational polynomial of degree \(n\) is expressible by radicals exactly when \(n \le 4\), the Gauss–Wantzel theorem on constructible regular polygons, and the essential dimension of \(S_n\) over \(\mathbb{C}\), which lies between \(\lfloor n/2 \rfloor\) and \(n - 3\) for \(n \ge 5\) (Buhler and Reichstein). Finite fields appear mostly through polynomials over them, such as the permutation polynomials, where \(X^d\) permutes \(\mathbb{F}_q\) exactly when \(\gcd(d, q - 1) = 1\), and field extensions and algebraic closure covers Puiseux's theorem on the algebraic closure of \(K((t))\) for \(K\) algebraically closed of characteristic zero.

• Galois theory
• Finite fields
• Field extensions and algebraic closure

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