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Algebraic number theory
Algebraic number theory
is a major branch of
number theory
that studies
algebraic structures
related to
algebraic integers
. This is generally accomplished by considering a
ring
of algebraic integers
O
in an
algebraic number field
K
/
Q
, and studying their algebraic properties such as
factorization
, the behaviour of
ideals
, and
field
extensions. In this setting, the familiar features of the
integers
—such as
unique factorization
—need not hold. The virtue of the primary machinery employed—
Galois theory
,
group cohomology
,
group representations
, and
L-functions
—is that it allows one to deal with new phenomena and yet partially recover the behaviour of the usual integers.
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