Last Updated：2025/11/27
(number theory) A p-adic absolute value, for a given prime number p, the function, denoted |..|ₚ and defined on the rational numbers, such that |0|ₚ = 0 and, for x≠0, |x|ₚ = p-ordₚ(x), where ordₚ(x) is the p-adic ordinal of x; the same function, extended to the p-adic numbers ℚₚ (the completion of the rational numbers with respect to the p-adic ultrametric defined by said absolute value); the same function, further extended to some extension of ℚₚ (for example, its algebraic closure).
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p-adic norm
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Source Word
p-adic norm
Noun
(number
theory)
A
p-adic
absolute
value,
for
a
given
prime
number
p,
the
function,
denoted
|..|ₚ
and
defined
on
the
rational
numbers,
such
that
|0|ₚ
=
0
and,
for
x≠0,
|x|ₚ
=
p^(-ordₚ(x)),
where
ordₚ(x)
is
the
p-adic
ordinal
of
x;
the
same
function,
extended
to
the
p-adic
numbers
ℚₚ
(the
completion
of
the
rational
numbers
with
respect
to
the
p-adic
ultrametric
defined
by
said
absolute
value);
the
same
function,
further
extended
to
some
extension
of
ℚₚ
(for
example,
its
algebraic
closure).
Japanese Meaning
（数論）素数pに対して定義されたp進絶対値。0の場合は0、非0の場合は|x|ₚ = p^(－ordₚ(x))となる。 / 有理数上に定義され、さらにp進数体ℚₚやその拡大体上へ拡張される関数。
Sense（1）
(number
theory)
A
p-adic
absolute
value,
for
a
given
prime
number
p,
the
function,
denoted
|..|ₚ
and
defined
on
the
rational
numbers,
such
that
|0|ₚ
=
0
and,
for
x≠0,
|x|ₚ
=
p^(-ordₚ(x)),
where
ordₚ(x)
is
the
p-adic
ordinal
of
x;
the
same
function,
extended
to
the
p-adic
numbers
ℚₚ
(the
completion
of
the
rational
numbers
with
respect
to
the
p-adic
ultrametric
defined
by
said
absolute
value);
the
same
function,
further
extended
to
some
extension
of
ℚₚ
(for
example,
its
algebraic
closure).
（ plural ）
