Last Updated
：2025/12/04
splitting field
Noun
of a K-algebra
of a central simple algebra
(algebra,
Galois
theory)
(of
a
polynomial)
Given
a
polynomial
p
over
a
field
K,
the
smallest
extension
field
L
of
K
such
that
p,
as
a
polynomial
over
L,
decomposes
into
linear
factors
(polynomials
of
degree
1);
(of
a
set
of
polynomials)
given
a
set
P
of
polynomials
over
K,
the
smallest
extension
field
of
K
over
which
every
polynomial
in
P
decomposes
into
linear
factors.
Japanese Meaning
与えられた多項式または多項式の集合が、1次多項式（一次式）に因数分解できるようになるよう、元の体の最小の拡大体。 / ガロア理論において、ある体上の多項式または多項式集合が全て1次因子に分解することができる、体の最小の拡大。
Sense（1）
(algebra,
Galois
theory)
(of
a
polynomial)
Given
a
polynomial
p
over
a
field
K,
the
smallest
extension
field
L
of
K
such
that
p,
as
a
polynomial
over
L,
decomposes
into
linear
factors
(polynomials
of
degree
1);
(of
a
set
of
polynomials)
given
a
set
P
of
polynomials
over
K,
the
smallest
extension
field
of
K
over
which
every
polynomial
in
P
decomposes
into
linear
factors.
Sense（2）
Sense（3）
（ plural ）
Quizzes for review
(algebra, Galois theory) (of a polynomial) Given a polynomial p over a field K, the smallest extension field L of K such that p, as a polynomial over L, decomposes into linear factors (polynomials of degree 1); (of a set of polynomials) given a set P of polynomials over K, the smallest extension field of K over which every polynomial in P decomposes into linear factors.
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See correct answer
splitting field
To determine the structure of the Galois group, we first compute the splitting field of the polynomial p over the base field K.
See correct answer
To determine the structure of the Galois group, we first compute the splitting field of the polynomial p over the base field K.
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English - English
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