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Constructing Subalgebras of K[x] Using the Minimal Polynomial

Ljunggren, Marianne LU (2023) In Master's Theses in Mathematical Sciences FMAM05 20231
Mathematics (Faculty of Engineering)
Abstract
In this report we will be working with subalgebras A of finite codimension in K[x]. It is known that such subalgebras can be expressed using a set of linear conditions evaluated at a finite set of points called the spectrum elements of A. These conditions are of one of two types, equality conditions or α-derivations, which in turn consists of the values and the values of the derivations of the elements in our algebra. From this representation we find a way to construct a polynomial, the zeros of which are exactly the spectrum element. This polynomial, called the minimal polynomial of A, has the property that its product with an arbitrary polynomial lies in our algebra. In order to find subalgebras of A we can add an additional condition,... (More)
In this report we will be working with subalgebras A of finite codimension in K[x]. It is known that such subalgebras can be expressed using a set of linear conditions evaluated at a finite set of points called the spectrum elements of A. These conditions are of one of two types, equality conditions or α-derivations, which in turn consists of the values and the values of the derivations of the elements in our algebra. From this representation we find a way to construct a polynomial, the zeros of which are exactly the spectrum element. This polynomial, called the minimal polynomial of A, has the property that its product with an arbitrary polynomial lies in our algebra. In order to find subalgebras of A we can add an additional condition, namely an α-derivation, where α lies in the spectrum of A. To find all such α-derivations, which can be written as a linear combination of regular derivations, we find an upper limit to the order of the derivations involved. To fully determine the derivation we also construct a method of finding all the required restrictions on the coefficients of said linear combination. (Less)
Popular Abstract
This work is about the mathematical structure algebra, where in our case the elements consists of polynomials. Unlike Ideals which have been well studied, subalgebras are less so. First we present a different way of expressing these algebras using only the values of polynomials and their derivatives at certain points. Given this representation this thesis contains a new method of finding subalgebras, which are still roughly the same size as the original algebra. This could potentially be useful when trying to analyse and classify these fairly unknown algebras further. A secondary result of the thesis is the method of constructing a particular polynomial such that the ideal generated by this polynomial lies entirely inside our algebra.
Please use this url to cite or link to this publication:
author
Ljunggren, Marianne LU
supervisor
organization
course
FMAM05 20231
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Algebra, Subalgebra, Polynomial Algebra, Minimal Polynomial
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3518-2023
ISSN
1404-6342
other publication id
2023:E51
language
English
id
9129000
date added to LUP
2024-03-05 13:27:49
date last changed
2024-03-05 13:27:49
@misc{9129000,
  abstract     = {{In this report we will be working with subalgebras A of finite codimension in K[x]. It is known that such subalgebras can be expressed using a set of linear conditions evaluated at a finite set of points called the spectrum elements of A. These conditions are of one of two types, equality conditions or α-derivations, which in turn consists of the values and the values of the derivations of the elements in our algebra. From this representation we find a way to construct a polynomial, the zeros of which are exactly the spectrum element. This polynomial, called the minimal polynomial of A, has the property that its product with an arbitrary polynomial lies in our algebra. In order to find subalgebras of A we can add an additional condition, namely an α-derivation, where α lies in the spectrum of A. To find all such α-derivations, which can be written as a linear combination of regular derivations, we find an upper limit to the order of the derivations involved. To fully determine the derivation we also construct a method of finding all the required restrictions on the coefficients of said linear combination.}},
  author       = {{Ljunggren, Marianne}},
  issn         = {{1404-6342}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Master's Theses in Mathematical Sciences}},
  title        = {{Constructing Subalgebras of K[x] Using the Minimal Polynomial}},
  year         = {{2023}},
}