Fundamental Constructs of Field Theory

• A field is a commutative division ring.

• The following axioms hold for a field (which follows from the properties of rings).

  We have operations (addition) and (multiplication)

  • Associativity of Addition:
  • Associativity of Multiplication:
  • Commutativity of Addition:
  • Commutativity of Multiplication:
  • Additive Identity: such that
  • Multiplicative Identity: such that
  • .
  • Additive Inverses such that
  • Multiplicative Inverses: such that .
  • Distributivity .

• (Fraleigh 19.9) Every field is an Integral Domain

• (Fraleigh 19.12) If is prime, then is a field

• A subfield of a field is a subset of the field that is afield under induced operations from the whole fiield. It is a generalization of Subgroup.

  • (Fraleigh e18.49b) The intersection of subfields of a field is again a subfield of
  • A field is an extension field of if .

• The following hold for ordered fields with positive set and the relation defined as

  • (Fraleigh e25.22)
  • (Fraleigh e25.23)
  • (Fraleigh e25.24) .

• (Fraleigh 27.6) A field has exactly ideals - and .

  • Proof: Clearly if we have the non-zero ideal, then let be the ideal and . Then

• (Fraleigh 27.11) A commutative ring with unity is a field if and only if it has no proper nontrivial ideals.

• An element of a field is an -th root of unity if . It is a primitive -th root of unity if and for .

Links §

• Fraleigh