Index

• algebra over $F$ §1.2
• algebraic §1.11, §1.12
• algebraic closure Definition 1.43
  • in an extension §1.14
• algebraic integer Aside 1.15
• algebraically closed Definition 1.43
• algorithm
  • division 1.6
  • Euclid’s 1.8
  • factoring a polynomial Remark 1.17
• automorphism §3.1
• characteristic
  • p §1.3
  • zero §1.3
• characteristic exponent §1.3
• commutative §1.1
• composite of fields §1.8
• conjugates Remark 3.11
• constructible §1.13, §3.5
• degree §1.6
  • separable Remark 3.14
• Eisenstein’s criterion Proposition 1.16
• element
  • separable Remark 3.14
• extension §1.6
  • abelian Definition 3.15
  • algebraic §1.11
  • cyclic Definition 3.15
  • finite §1.6
  • Galois Definition 3.10
  • inseparable Definition 3.6
  • normal Definition 3.7
  • separable Definition 3.6
  • simple §1.8
  • solvable Definition 3.15
  • transcendental §1.11
• $F$-algebra §1.2
• $F$-isomorphism §2.1
• $F$-homomorphism §1.6
• field Definition 1.1
  • perfect Definition 2.15
  • prime §1.3
  • stem §1.10
• fixed field §3.1
• $F^{q}$ §1.3
• Frobenius
  • endomorphism item a
• Frobenius endomorphism 1.4
• fundamental theorem
  • of algebra Remark 1.17, Proof.
• Galois closure Remark 3.17
• Galois group Definition 3.10
  • of a polynomial §3.6
• Gaussian numbers Example 1.19
• $\gcd$ §1.1
• group
  • Cremona Example 3.1
• homomorphism
  • of $F$-algebras §1.2
  • of fields §1.2
  • of rings §1.1
• ideal §1.1
• integral domain §1.1
• invariants §3.1
• Lemma
  • Gauss’s Proposition 1.13
• multiplicity §2.3
• normal closure Remark 3.17
• PARI Remark 1.17, Remark 1.17, Example 1.27, Remark 1.29, 1.8
• perfect field Definition 2.15
• polynomial
  • minimal §1.11
  • minimum §1.11
  • monic 1.9
  • separable Definition 2.14
• prime
  • Fermat §1.13
• ring §1.1
• root
  • multiple §2.3
  • of a polynomial 1.7
  • simple §2.3
• $S_{n}$ §3.6
• solvable in radicals §3.7
• split §1.14, §2.2
• splitting field §2.2
• subfield §1.2
  • generated by subset §1.8
• subring §1.1
  • generated by subset §1.7
• theorem
  • Artin’s Theorem 3.4
  • binomial in characteristic $p$ 1.4
  • constructible numbers Theorem 1.36, Theorem 3.23
  • fundamental of algebra §1.14, §1.14
  • fundamental of Galois theory Theorem 3.16
  • Galois 1832 Theorem 3.27
  • Galois extensions Theorem 3.9
  • Liouville Theorem 1.33
• transcendental §1.11, §1.12