Algebraic Number Theory Background (1)

Number Fields

A number field can be defined as a field extension  obtained by adjoining an abstract element  to the field of rationals, where  satisfies the relation  for some irreducible polynomial  which is monic without loss of generality. The polynomial  is called the minimal polynomial of  and the degree  of the number field is the degree of .

Because  the number field  can be seen as an  -dimensional vector space over  with basis  this is called the power basis of . Of course, associating  with indeterminate  yields a natural isomorphism between  and .

Let  be a positive integer, and let  denote an element of multiplicative order  i.e., a primitive th root of unity. The th cyclotomic number field is  and the minimal polynomial of  is the th cyclotomic polynomial

where  is any primitive th complex root of unity, e.g.,  Observe that the complex roots  of  are exactly the primitive  th roots of unity in  and that  has degree  the totient of 

Embeddings and Geometry

Here we describe the embeddings of a number field, which induce a natural 'canonical' geometry on it.

A number field  of degree  has exactly  ring embeddings (injective ring homomorphisms)  Concretely, these embeddings map  to each of the complex roots of its minimal polynomial ; it is easy to see that these are the only ring embeddings from  to  because  An embedding whose image lies in  (corresponding to a real root of  ) is called a real embedding; otherwise (for a complex root of  ) it is called a complex embedding. Because complex roots of  come in conjugate pairs, so too do the complex embeddings. The number of real embeddings is denoted  and the number of pairs of complex embeddings is denoted  so we have  By convention, we let  be the real embeddings, and we order the complex embeddings so that  for  The canonicalembedding  is then defined as

By identifying elements of  with their canonical embeddings in  we can speak of geometric norms (e.g., the Euclidean norm) on . Recalling that we define norms on  as those induced from , we see that for any  and any , the  norm of  is simply  for  and is  for  (As always, we assume the  norm when  is omitted.) Because multiplication of embedded elements is component-wise (since  is a ring homomorphism), we have

for any  and any  Thus the  norm acts as an 'absolute value' for  that bounds how much an element 'expands' any other by multiplication.

Using the canonical embedding also allows us to think of the Gaussian distribution  for  over  or its discrete analogue over a lattice in  as a distribution over  Strictly speaking, the distribution  is not over  but rather over the field tensor product  which is isomorphic to  Since multiplication of elements in  is mapped to coordinate-wise multiplication in  we have that for any element  the distribution of  is  where  (This uses the fact that our distributions have the same variance in the real and imaginary components of each complex embedding.)

• Example
  For the  th cyclotomic field where  for  there are  complex embeddings (and no real ones), which are given by  for  (In this case it is convenient to index the embeddings  by elements of  instead of  ) For any power  all the embeddings are roots of unity and hence have magnitude  so  and