Algebraic closure roadmap

Roadmap to algebraic closure of field F

Implicitly, in what I write below, p=1 if char(F)=0.

Field theory

• 1. Any finite subgroup of F* is cyclic
• 2. Freshman's dream: (x+y)^p = x^p + y^p

Polynomials

• 3. division algorithm
• 4. F[X] is a ED (3)
• 5. ED -> PID -> UFD -> GCD
• 6. F[X] is a PID (4,5)
• 7. F[X] is a UFD (4,5)
• 8. F[X] is a GCD (4,5)

Algebraic extensions

• 9. Integral elements form subring [currently being PRd]

Splitting fields

• 10. Every polynomial splits in some extension (6,7) [on the splitting_fields branch]

Separable polynomials

• 11. Resultant of two polynomials
• 12. Discriminant of polynomials (11)
• 13. f in F[X] is separable := discriminant(f) is non-zero (12)
• 14. f is separable iff gcd(f,Df)=1 (8,11)
• 14'. Alternative: take (14) as definition of separable.
• 15. f is separable iff it has no double root in every extension in which it splits (10)
• 16. For every irreducible f in F[X] there is n in N and h in F[X] such that f(x) = h(x^(p^n)) and h is separable (14)
• 17. If If K/F then f in F[X] is separable iff (coe f) in K[X] is separable. (13)
• 18. f is separable iff all its factors are separable
• 19. Primitive element theorem (1,10,15,17,18)
• 20. F perfect := every polynomial is separable
• 21. Perfect iff Frobenius surjective (2,16)

Proof

• 22. For every irreducible f in F[X] let Xf be an indeterminate.
• 23. Let R := F[{Xf | f irredcuible}] be a big polynomial ring.
• 24. Let I := span {f(Xf) | f irreducible} an ideal in R.
• 25. I is a proper ideal. (10)
• 26. Let L := R/M where M is a maximal ideal that contains I (25).
• 27. L is a field algebraic over F. (9)
• 28. Let K := {x in L | exists n, x^(p^n) in F}.
• 29. K is a subfield. (2)
• 30. K is perfect. (2,21)
• 31. Every polynomial in K[X] has a root in L. (2)
• 32. Every polynomial in K[X] splits in L. (19,30,31)
• 33. L is algebraically closed. (32)
• 34. L is an algebraic closure of F. (27,33)