Tschirnhaus transformation
Encyclopedia
In mathematics
, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomial
s developed by Ehrenfried Walther von Tschirnhaus
in 1683. It may be defined conveniently by means of field theory
, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial
that takes a root to some rational function
applied to that root.
In detail, let K be a field, and P(t) a polynomial over K. If P is irreducible, then
the quotient ring
of the polynomial ring
K[t] by the principal ideal
generated by P, is a field extension
of K. We have
where α is t modulo (P). That is, α is a primitive element of L. There will be other choices β of primitive element in L: for any such choice of β we will have
with polynomials F and G over K. In fact this follows from the quotient representation above. Now if Q is the minimal polynomial for β over K, we can call Q a Tschirnhaus transformation of P.
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing P, but leaving L the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory
, when L is a Galois extension
of K. The Galois group
is then described (in one way) as all the Tschirnhaus transformations of P to itself.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Tschirnhaus transformation, also known as Tschirnhausen transformation, is a type of mapping on polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...
s developed by Ehrenfried Walther von Tschirnhaus
Ehrenfried Walther von Tschirnhaus
Ehrenfried Walther von Tschirnhaus was a German mathematician, physicist, physician, and philosopher...
in 1683. It may be defined conveniently by means of field theory
Field theory (mathematics)
Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
, as the transformation on minimal polynomials implied by a different choice of primitive element. This is the most general transformation of an irreducible polynomial
Irreducible polynomial
In mathematics, the adjective irreducible means that an object cannot be expressed as the product of two or more non-trivial factors in a given set. See also factorization....
that takes a root to some rational function
Rational function
In mathematics, a rational function is any function which can be written as the ratio of two polynomial functions. Neither the coefficients of the polynomials nor the values taken by the function are necessarily rational.-Definitions:...
applied to that root.
In detail, let K be a field, and P(t) a polynomial over K. If P is irreducible, then
- K[t]/(P(t)) = L,
the quotient ring
Quotient ring
In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of the polynomial ring
Polynomial ring
In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
K[t] by the principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
generated by P, is a field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
of K. We have
- L = K(α)
where α is t modulo (P). That is, α is a primitive element of L. There will be other choices β of primitive element in L: for any such choice of β we will have
- β = F(α), α = G(β),
with polynomials F and G over K. In fact this follows from the quotient representation above. Now if Q is the minimal polynomial for β over K, we can call Q a Tschirnhaus transformation of P.
Therefore the set of all Tschirnhaus transformations of an irreducible polynomial is to be described as running over all ways of changing P, but leaving L the same. This concept is used in reducing quintics to Bring–Jerrard form, for example. There is a connection with Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
, when L is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
of K. The Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
is then described (in one way) as all the Tschirnhaus transformations of P to itself.