By Stein W.A.

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Nr ) = 1, though it implies this. 1) that sends a ∈ Z to its reduction modulo each ni , is an isomorphism. This map is never an isomorphism if the ni are not coprime. 1) is lcm(n1 , . . , nr ), since it is the image of a cyclic group and lcm(n1 , . . , nr ) is the largest order of an element of the right hand side, whereas the cardinality of the right hand side is n1 · · · nr . , x ≡ ar (mod nr ) with pairwise coprime moduli has a unique solution modulo n1 · · · nr . 61 62 CHAPTER 5. 1). There is a natural map φ : Z → (Z/n1 Z) ⊕ · · · ⊕ (Z/nr Z) given by projection onto each factor.

30 CHAPTER 2. 12 (Number field). A number field is a field K that contains the rational numbers Q such that the degree [K : Q] = dimQ (K) is finite. If K is a number field, then by the primitive element theorem there is an α ∈ K so that K = Q(α). Let f (x) ∈ Q[x] be the minimal polynomial of α. Fix a choice of algebraic closure Q of Q. Associated to each of the deg(f ) roots α ∈ Q of f , we obtain a field embedding K → Q that sends α to α . Thus any number field can be embedded in [K : Q] = deg(f ) distinct ways in Q.

Here “coprime in pairs” means that gcd(ni , nj ) = 1 whenever i = j; it does not mean that gcd(n1 , . . , nr ) = 1, though it implies this. 1) that sends a ∈ Z to its reduction modulo each ni , is an isomorphism. This map is never an isomorphism if the ni are not coprime. 1) is lcm(n1 , . . , nr ), since it is the image of a cyclic group and lcm(n1 , . . , nr ) is the largest order of an element of the right hand side, whereas the cardinality of the right hand side is n1 · · · nr . , x ≡ ar (mod nr ) with pairwise coprime moduli has a unique solution modulo n1 · · · nr .