Pmath 441/641 Algebraic Number Theory

Definition. An algebraic integer is the root of a monic polynomial in Z[x]. An algebraic number is the root of any non-zero polynomial in Z[x]. We are interested in studying the structure of the ring of algebraic integers in an algebraic number field. A number field is a finite extension of Q. We’ll assume that the number fields we consider are all subfields of C. Definition. Suppose that K and L are fields with K ⊆ L. Then K is a subfield of L and L is an extension field of K. We denote the dimension of L as a vector space over K by [L : K] . If [L : K] <∞, we say L is a finite extension of K. Definition. Suppose that H is a field with K ⊆ H ⊆ L. Then we say H is an intermediate field of K and L. Recall that [L : K] = [L : H][H : K]. Definition. A polynomial f ∈ K[x] is said to be irreducible over K iff whenever f = gh with g, h ∈ K[x], we have g or h constant. Recall that K[x] is a Principal Ideal Domain. Definition. Let K be a subfield of C and let θ ∈ C be an algebraic number. We denote by K(θ) the smallest subfield of C containing K and θ, Definition. Let K be a subfield of C and let θ ∈ C to be algebraic over K. A polynomial in K[x] is said to be a minimal polynomial of θ over K if it is monic, has θ as a root, and has degree as small as possible with these properties.