Constructing Subalgebras of K[x] Using the Minimal Polynomial
(2023) In Master's Theses in Mathematical Sciences FMAM05 20231Mathematics (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.
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http://lup.lub.lu.se/student-papers/record/9129000
- author
- Ljunggren, Marianne LU
- supervisor
- organization
- course
- FMAM05 20231
- year
- 2023
- 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}}, }