In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
and more specifically in
field theory, a radical extension of a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''K'' is an
extension
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (predicate logic), the set of tuples of values that satisfy the predicate
* E ...
of ''K'' that is obtained by adjoining a sequence of
''n''th roots of elements.
Definition
A simple radical extension is a
simple extension In field theory, a simple extension is a field extension which is generated by the adjunction of a single element. Simple extensions are well understood and can be completely classified.
The primitive element theorem provides a characterization ...
''F''/''K'' generated by a single element
satisfying
for an element ''b'' of ''K''. In
characteristic ''p'', we also take an extension by a root of an
Artin–Schreier polynomial to be a simple radical extension. A radical series is a
tower
A tower is a tall Nonbuilding structure, structure, taller than it is wide, often by a significant factor. Towers are distinguished from guyed mast, masts by their lack of guy-wires and are therefore, along with tall buildings, self-supporting ...
where each extension
is a simple radical extension.
Properties
# If ''E'' is a radical extension of ''F'' and ''F'' is a radical extension of ''K'' then ''E'' is a radical extension of ''K''.
# If ''E'' and ''F'' are radical extensions of ''K'' in an
extension field
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ' ...
''C'' of ''K'', then the
compositum
In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subf ...
''EF'' (the smallest subfield of ''C'' that contains both ''E'' and ''F'') is a radical extension of ''K''.
# If ''E'' is a radical extension of ''F'' and ''E'' > ''K'' > ''F'' then ''E'' is a radical extension of ''K''.
Solvability by radicals
Radical extensions occur naturally when solving
polynomial equation
In mathematics, an algebraic equation or polynomial equation is an equation of the form
:P = 0
where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation' ...
s in
radicals. In fact a
solution in radicals
A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divi ...
is the expression of the solution as an element of a radical series: a polynomial ''f'' over a field ''K'' is said to be solvable by radicals if there is a
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
of ''f'' over ''K'' contained in a radical extension of ''K''.
The
Abel–Ruffini theorem
In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means th ...
states that such a solution by radicals does not exist, in general, for equations of degree at least five.
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radicals, ...
showed that an equation is solvable in radicals if and only if its
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
is
solvable. The proof is based on the
fundamental theorem of Galois theory
In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions in relation to groups. It was proved by Évariste Galois in his development of Galois theory.
In its most basi ...
and the following theorem.
The proof is related to
Lagrange resolvent
In Galois theory, a discipline within the field of abstract algebra, a resolvent for a permutation group ''G'' is a polynomial whose coefficients depend polynomially on the coefficients of a given polynomial ''p'' and has, roughly speaking, a rati ...
s. Let
be a
primitive ''n''th root of unity (belonging to ''K''). If the extension is generated by
with
as a
minimal polynomial, the mapping
induces a ''K''-automorphism of the extension that generates the Galois group, showing the "only if" implication. Conversely, if
is a ''K''-automorphism generating the Galois group, and
is a generator of the extension, let
:
The relation
implies that the product of the
conjugates of
(that is the images of
by the ''K''-automorphisms) belongs to ''K'', and is equal to the product of
by the product of the ''n''th roots of unit. As the product of the ''n''th roots of units is
, this implies that
and thus that the extension is a radical extension.
It follows from this theorem that a Galois extension may be extended to a radical extension if and only if its Galois group is solvable (but there are non-radical Galois extensions whose Galois group is solvable, for example
). This is, in modern terminology, the criterion of solvability by radicals that was provided by Galois. The proof uses the fact that the
Galois closure
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
of a simple radical extension of degree ''n'' is the extension of it by a primitive ''n''th root of unity, and that the Galois group of the ''n''th roots of unity is cyclic.
References
*
* {{cite book , last=Roman , first=Steven , title=Field theory , edition=2nd , zbl=1172.12001 , series=Graduate Texts in Mathematics , volume=158 , location=New York, NY , publisher=
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
, isbn=0-387-27677-7 , year=2006
Galois theory
Equations