The tree of primes in a field
The product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. We propose a new concept of prime for any field and investigate some of its properties. There are algebraic primes, co...
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| Lenguaje: | English |
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Universidad de La Frontera. Departamento de Matemática y Estadística.
2010
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oai:scielo:S0719-064620100002000072018-10-08The tree of primes in a fieldRump,Wolfgang prime valuation product formula The product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. We propose a new concept of prime for any field and investigate some of its properties. There are algebraic primes, corresponding to valuations, such that every prime contains a largest algebraic one. For a number field, this algebraic part is zero just for the infinite primes. It is shown that the primes of any field form a tree with a kind of self-similar structure, and there is a binary operation on the primes, unexplored even for the rationals. Every prime defines a topology on the field, and each compact prime gives rise to a unique Haar measure, playing an essential part in the product formula.info:eu-repo/semantics/openAccessUniversidad de La Frontera. Departamento de Matemática y Estadística.Cubo (Temuco) v.12 n.2 20102010-01-01text/htmlhttp://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200007en10.4067/S0719-06462010000200007 |
| institution |
Scielo Chile |
| collection |
Scielo Chile |
| language |
English |
| topic |
prime valuation product formula |
| spellingShingle |
prime valuation product formula Rump,Wolfgang The tree of primes in a field |
| description |
The product formula of algebraic number theory connects finite and infinite primes in a stringent way, a fact, while not hard to be checked, that has never ceased to be tantalizing. We propose a new concept of prime for any field and investigate some of its properties. There are algebraic primes, corresponding to valuations, such that every prime contains a largest algebraic one. For a number field, this algebraic part is zero just for the infinite primes. It is shown that the primes of any field form a tree with a kind of self-similar structure, and there is a binary operation on the primes, unexplored even for the rationals. Every prime defines a topology on the field, and each compact prime gives rise to a unique Haar measure, playing an essential part in the product formula. |
| author |
Rump,Wolfgang |
| author_facet |
Rump,Wolfgang |
| author_sort |
Rump,Wolfgang |
| title |
The tree of primes in a field |
| title_short |
The tree of primes in a field |
| title_full |
The tree of primes in a field |
| title_fullStr |
The tree of primes in a field |
| title_full_unstemmed |
The tree of primes in a field |
| title_sort |
tree of primes in a field |
| publisher |
Universidad de La Frontera. Departamento de Matemática y Estadística. |
| publishDate |
2010 |
| url |
http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462010000200007 |
| work_keys_str_mv |
AT rumpwolfgang thetreeofprimesinafield AT rumpwolfgang treeofprimesinafield |
| _version_ |
1714206765859471360 |