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gptkbp:instanceOf
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gptkb:mathematical_concept
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gptkbp:application
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algebraic number theory
cryptography
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gptkbp:defines
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A quadratic field is a number field of the form Q(√d) where d is a square-free integer.
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gptkbp:degree
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2
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gptkbp:example
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Q(√5)
Q(√-1)
Q(√-3)
Q(√2)
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gptkbp:field
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algebraic number theory
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gptkbp:hasClassNumber
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integer invariant
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gptkbp:hasDiscriminant
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d or 4d depending on d mod 4
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gptkbp:hasRingOfIntegers
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Z[(1+√d)/2] or Z[√d]
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gptkbp:hasUnitGroup
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finite for imaginary, infinite for real quadratic fields
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https://www.w3.org/2000/01/rdf-schema#label
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quadratic fields
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gptkbp:imaginaryQuadraticField
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Q(√d) with d<0
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gptkbp:isA
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number field
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gptkbp:realQuadraticField
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Q(√d) with d>0
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gptkbp:relatedTo
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gptkb:Dedekind_zeta_function
gptkb:Dirichlet's_unit_theorem
gptkb:Hilbert_class_field
gptkb:Kronecker–Weber_theorem
quadratic forms
ideal class group
class number problem
unique factorization
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gptkbp:studiedBy
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gptkb:Carl_Friedrich_Gauss
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gptkbp:subclassOf
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algebraic number field
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gptkbp:bfsParent
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gptkb:Algebraic_number_theory
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gptkbp:bfsLayer
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5
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