Mathematische Annalen

, Volume 148, Issue 1, pp 31–64 | Cite as

On the theory of orders, in particular on the semigroup of ideal classes and genera of an order in an algebraic number field

  • E. C. Dade
  • O. Taussky
  • H. Zassenhaus


Number Field Algebraic Number Ideal Classis Algebraic Number Field 
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an integral domain

I (D)

the set of the fractional ideals ofD formed in the quotient field ofD, closed under the four operations ., +, :, ∩

S (D)

the multiplicative semigroup with division of the arithmetical equivalence classes ofI (D)


a noetherian ring

T (\(\mathfrak{O}\))

the multiplicative semigroup with division of the weak equivalence classes of the fractional ideals of\(\mathfrak{O}\)

G (\(\mathfrak{O}\))

the multiplicative group of all invertible\(\mathfrak{O}\)-ideals with order\(\mathfrak{O}\)



I (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the family of all ideals\(\mathfrak{a}\) ofI(\(\left( \mathfrak{O} \right)\)) such that\(\mathfrak{a} \mathfrak{O}' \in G\left( {\mathfrak{O}'} \right)\)G(\(\mathfrak{a} \mathfrak{O}' \in G\left( {\mathfrak{O}'} \right)\))

T (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the family of weak equivalence classes of elements ofI (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

J (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the family of all\(\mathfrak{O}\)-submodules ≠ 0 of\(\mathfrak{O}'\)

U (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the family of all ideals\(\mathfrak{a}\) ofJ (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\)) such that\(\mathfrak{a}\mathfrak{O}' = \mathfrak{O}'\)

V (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the multiplicative semigroup with division of the weak equivalence classes contained inU (\(\left( {{{\mathfrak{O}'} \mathord{\left/ {\vphantom {{\mathfrak{O}'} \mathfrak{O}}} \right. \kern-\nulldelimiterspace} \mathfrak{O}}} \right)\))

the ring of the rational integers


the rational field

\(\mathfrak{O}_\mathfrak{p} \)

the local ring of all elementsx/ν (x\(\mathfrak{O}\),ν\(\mathfrak{O}\),ν\(\mathfrak{p}\)) belonging to the prime ideal\(\mathfrak{p}\) of the integral domain\(\mathfrak{O}\)

\(\mathfrak{a}_\mathfrak{p} \)

the local extension a\(\mathfrak{a} \mathfrak{O}_\mathfrak{p} \) of a member\(\mathfrak{a}\) ofI (\(\mathfrak{O}\)) to a member ofI (\(\mathfrak{O}_\mathfrak{p} \))

N (\(\mathfrak{a}\))

a non-negative integer with the property that\(\mathfrak{a}^{N\left( \mathfrak{a} \right)} \) is invertible, but\(\mathfrak{a}^{N\left( \mathfrak{a} \right) - 1} \) is not invertible.


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Copyright information

© Springer-Verlag 1962

Authors and Affiliations

  • E. C. Dade
    • 1
  • O. Taussky
    • 2
  • H. Zassenhaus
    • 3
  1. 1.Pasadena
  2. 2.Pasadena
  3. 3.Notre Dame

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