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Splitting field
Meaning
Noun
●
In Algebra:
(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
l
inear 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
l
inear factors.
●
In Algebra:
Given a
finite-dimensional
K-algebra (
algebra over a field
), an
extension field
whose every
simple
(indecomposable)
module
is
absolutely
simple
(remains
simple
after the
scalar field
has been
extended
to said
extension field
).
●
In Algebra:
Given a
central simple algebra
A over a
field
K, another
field
, E,
such that
the
tensor product
A⊗E is
isomorphic
to a
matrix
ring
over E.
●
In Algebra:
(of a
character
χ of a
representation
of a
g
roup G) A
field
K over which a K-representation of G
exists
which includes the
character
χ; (of a
g
roup G) a
field
over which a K-representation of G
exists
which includes every
irreducible
character
in G.
Sourced from
Wiktionary
Synonyms
Root Field
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