Understanding the Tschirnhaus transformation The Next CEO of Stack OverflowSplitting field and dimension of irreducible polynomialsComputing the Galois group of $x^4+ax^2+b in mathbbQ[x] $Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem?Galois group. $K$-automorphisms take adjoined roots to other roots of minimal polynomial or take roots of $f$ to other roots of $f$Degree of the field extensionminimal polynomial of $alpha - beta$ from the minimal polynomial of $alpha$Is there a way I can find out the degree of this extension without explicitly finding the minimal polynomial?Understanding this Abstract Algebra TheoremNumber of elements of a splitting fieldCharacterising the irreducible polynomials in positive characteristic whose roots generate the (cyclic) group of units of the splitting field
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Understanding the Tschirnhaus transformation
The Next CEO of Stack OverflowSplitting field and dimension of irreducible polynomialsComputing the Galois group of $x^4+ax^2+b in mathbbQ[x] $Existence of irreducible polynomial of arbitrary degree over finite field without use of primitive element theorem?Galois group. $K$-automorphisms take adjoined roots to other roots of minimal polynomial or take roots of $f$ to other roots of $f$Degree of the field extensionminimal polynomial of $alpha - beta$ from the minimal polynomial of $alpha$Is there a way I can find out the degree of this extension without explicitly finding the minimal polynomial?Understanding this Abstract Algebra TheoremNumber of elements of a splitting fieldCharacterising the irreducible polynomials in positive characteristic whose roots generate the (cyclic) group of units of the splitting field
$begingroup$
I would like to understand the definition of a Tschrinhaus transformation.
Take the irreducible polynomial $p(X) = 2X^2 + 2 in mathbbF_3[X]$ and define $L := mathbbF_3[X]/(p(X)) cong mathbbF_9$. So $L$ is a finite extension of $mathbbF_3$.
Now Wikipedia states that $L = mathbbF_3(alpha)$ with $alpha = X text mod p(X)$ and the task is to find different primitive elements $beta$ such that $L = mathbbF_3(beta)$. Then the minimal polynomial of $beta$ is called a Tschirnhaus transformation.
My question is: How to find the different primitive elements? Does this depends on the specific case? Can I just use all irreducible polynomials $q(X)$ of $mathbbF_3[X]$ with $deg(q(X)) leq deg(p(X))$?
abstract-algebra polynomials galois-theory
asked Mar 28 at 10:38
SqyuliSqyuli
344111
$endgroup$
add a comment |
$begingroup$
I would like to understand the definition of a Tschrinhaus transformation.
Take the irreducible polynomial $p(X) = 2X^2 + 2 in mathbbF_3[X]$ and define $L := mathbbF_3[X]/(p(X)) cong mathbbF_9$. So $L$ is a finite extension of $mathbbF_3$.
Now Wikipedia states that $L = mathbbF_3(alpha)$ with $alpha = X text mod p(X)$ and the task is to find different primitive elements $beta$ such that $L = mathbbF_3(beta)$. Then the minimal polynomial of $beta$ is called a Tschirnhaus transformation.
My question is: How to find the different primitive elements? Does this depends on the specific case? Can I just use all irreducible polynomials $q(X)$ of $mathbbF_3[X]$ with $deg(q(X)) leq deg(p(X))$?
abstract-algebra polynomials galois-theory
asked Mar 28 at 10:38
SqyuliSqyuli
344111
$endgroup$
add a comment |
$begingroup$
I would like to understand the definition of a Tschrinhaus transformation.
Take the irreducible polynomial $p(X) = 2X^2 + 2 in mathbbF_3[X]$ and define $L := mathbbF_3[X]/(p(X)) cong mathbbF_9$. So $L$ is a finite extension of $mathbbF_3$.
Now Wikipedia states that $L = mathbbF_3(alpha)$ with $alpha = X text mod p(X)$ and the task is to find different primitive elements $beta$ such that $L = mathbbF_3(beta)$. Then the minimal polynomial of $beta$ is called a Tschirnhaus transformation.
My question is: How to find the different primitive elements? Does this depends on the specific case? Can I just use all irreducible polynomials $q(X)$ of $mathbbF_3[X]$ with $deg(q(X)) leq deg(p(X))$?
abstract-algebra polynomials galois-theory
asked Mar 28 at 10:38
SqyuliSqyuli
344111
$endgroup$
I would like to understand the definition of a Tschrinhaus transformation.
Take the irreducible polynomial $p(X) = 2X^2 + 2 in mathbbF_3[X]$ and define $L := mathbbF_3[X]/(p(X)) cong mathbbF_9$. So $L$ is a finite extension of $mathbbF_3$.
Now Wikipedia states that $L = mathbbF_3(alpha)$ with $alpha = X text mod p(X)$ and the task is to find different primitive elements $beta$ such that $L = mathbbF_3(beta)$. Then the minimal polynomial of $beta$ is called a Tschirnhaus transformation.
My question is: How to find the different primitive elements? Does this depends on the specific case? Can I just use all irreducible polynomials $q(X)$ of $mathbbF_3[X]$ with $deg(q(X)) leq deg(p(X))$?
abstract-algebra polynomials galois-theory
abstract-algebra polynomials galois-theory
asked Mar 28 at 10:38
SqyuliSqyuli
344111
asked Mar 28 at 10:38
SqyuliSqyuli
344111
edited Mar 28 at 11:43
Sqyuli
asked Mar 28 at 10:38
SqyuliSqyuli
344111
asked Mar 28 at 10:38
SqyuliSqyuli
344111
asked Mar 28 at 10:38
SqyuliSqyuli
344111
344111
add a comment |
add a comment |
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