| Notes on Geometry and 3-Manifolds (2007) | |||||||||||||||
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| ly, any number field is isomorphic to a field of the form K = Q(x)=(f(x)), where f(x) 2 Q[x] is an irreducible poynomial. If is a root of the polynomial f(x) then K embeds in C by the map induced by x 7! . Conversely, any embedding K ! C is obtained this way, so the complex embeddings of K are in 1-1 correspondence with the roots of f(x). Such an embedding is called a real embedding if its image lies in R, i.e., if the corresponding root of f(x) is real. 31 32 3. ARITHMETIC INVARIANTS If we denote by r 1 the number of real embeddings of K (real roots of f(x)) and r 2 the number of conjugate pairs of complex embeddings (conjugate pairs of complex roots of f(x)) then we see that r 1 + 2r 2 = d, the degree of the extension K=Q (this is also the degree of the polynomial f(x). An element of C is an algebraic integer if it satisfies a monic polynomial equation with integral coefficients. The algebraic integers in a number field K form a subring OK of K called the ring of integers of K... | |||||||||||||||
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