Reference for Dedekind's Example of a Non-monogenic Field

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












An oft quoted fact is that Dedekind discovered that adjoining a root of $x^3-x^2-2x-8$ to $mathbbQ$ yields a number field that is not monogenic. Does anyone know exactly where Dedekind writes this? In other words, does anyone have the citation for this?




number-theory math-history




share|cite|improve this question



















  • I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
    – Lubin
    Jul 14 at 20:22










  • Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
    – DeerintheHeadlights
    Jul 15 at 2:39














up vote
2
down vote

favorite












An oft quoted fact is that Dedekind discovered that adjoining a root of $x^3-x^2-2x-8$ to $mathbbQ$ yields a number field that is not monogenic. Does anyone know exactly where Dedekind writes this? In other words, does anyone have the citation for this?




number-theory math-history




share|cite|improve this question



















  • I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
    – Lubin
    Jul 14 at 20:22










  • Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
    – DeerintheHeadlights
    Jul 15 at 2:39












up vote
2
down vote

favorite









up vote
2
down vote

favorite











An oft quoted fact is that Dedekind discovered that adjoining a root of $x^3-x^2-2x-8$ to $mathbbQ$ yields a number field that is not monogenic. Does anyone know exactly where Dedekind writes this? In other words, does anyone have the citation for this?




number-theory math-history




share|cite|improve this question











An oft quoted fact is that Dedekind discovered that adjoining a root of $x^3-x^2-2x-8$ to $mathbbQ$ yields a number field that is not monogenic. Does anyone know exactly where Dedekind writes this? In other words, does anyone have the citation for this?





number-theory math-history





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 14 at 16:28









DeerintheHeadlights

1317




1317











  • I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
    – Lubin
    Jul 14 at 20:22










  • Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
    – DeerintheHeadlights
    Jul 15 at 2:39
















  • I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
    – Lubin
    Jul 14 at 20:22










  • Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
    – DeerintheHeadlights
    Jul 15 at 2:39















I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
– Lubin
Jul 14 at 20:22




I presume you mean that the integers of the field can not be generated over $Bbb Z$ by a single element.
– Lubin
Jul 14 at 20:22












Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
– DeerintheHeadlights
Jul 15 at 2:39




Yes, by monogenic we mean that the ring of integers admits a $mathbbZ$-basis of the form $1,theta,theta^2,dots, theta^n-1$.
– DeerintheHeadlights
Jul 15 at 2:39










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










It can be found e.g. in his announcement of the second edition of Dirichlet's Lectures in Number Theory (Gött. gelehrte Anzeigen 1871, 1481--1494; see Dedekind Werke, vol III, p. 406). He published the details in Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Gött. Abhandlungen 1878, 1--23; Werke II, 202-223; see in particularp. 225.






share|cite|improve this answer





















  • Thank you very much!
    – DeerintheHeadlights
    Jul 15 at 21:42










Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);








 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2851742%2freference-for-dedekinds-example-of-a-non-monogenic-field%23new-answer', 'question_page');

);

Post as a guest

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
1
down vote



accepted










It can be found e.g. in his announcement of the second edition of Dirichlet's Lectures in Number Theory (Gött. gelehrte Anzeigen 1871, 1481--1494; see Dedekind Werke, vol III, p. 406). He published the details in Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Gött. Abhandlungen 1878, 1--23; Werke II, 202-223; see in particularp. 225.






share|cite|improve this answer





















  • Thank you very much!
    – DeerintheHeadlights
    Jul 15 at 21:42














up vote
1
down vote



accepted










It can be found e.g. in his announcement of the second edition of Dirichlet's Lectures in Number Theory (Gött. gelehrte Anzeigen 1871, 1481--1494; see Dedekind Werke, vol III, p. 406). He published the details in Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Gött. Abhandlungen 1878, 1--23; Werke II, 202-223; see in particularp. 225.






share|cite|improve this answer





















  • Thank you very much!
    – DeerintheHeadlights
    Jul 15 at 21:42












up vote
1
down vote



accepted







up vote
1
down vote



accepted






It can be found e.g. in his announcement of the second edition of Dirichlet's Lectures in Number Theory (Gött. gelehrte Anzeigen 1871, 1481--1494; see Dedekind Werke, vol III, p. 406). He published the details in Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Gött. Abhandlungen 1878, 1--23; Werke II, 202-223; see in particularp. 225.






share|cite|improve this answer













It can be found e.g. in his announcement of the second edition of Dirichlet's Lectures in Number Theory (Gött. gelehrte Anzeigen 1871, 1481--1494; see Dedekind Werke, vol III, p. 406). He published the details in Über den Zusammenhang zwischen der Theorie der Ideale und der Theorie der höheren Kongruenzen, Gött. Abhandlungen 1878, 1--23; Werke II, 202-223; see in particularp. 225.







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 15 at 8:49









franz lemmermeyer

6,31621742




6,31621742











  • Thank you very much!
    – DeerintheHeadlights
    Jul 15 at 21:42
















  • Thank you very much!
    – DeerintheHeadlights
    Jul 15 at 21:42















Thank you very much!
– DeerintheHeadlights
Jul 15 at 21:42




Thank you very much!
– DeerintheHeadlights
Jul 15 at 21:42












 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2851742%2freference-for-dedekinds-example-of-a-non-monogenic-field%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

What is the equation of a 3D cone with generalised tilt?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite What is the equation of a 3D cone with generalized tilt? I've noticed that in most equations given to represent a cone, there is no parameter which defines the tilt of the cone in 3D space and that most of them have their point at the origin $(0,0,0)$ - I was wondering if anyone could give me a more generalized cone equation for a cone in any position in 3D space 3d share | cite | improve this question edited Jul 25 '16 at 0:05 ervx 9,425 3 13 36 asked Jul 24 '16 at 15:38 Charlene 11 2 2 Welcome to Math.SE! Could you please explain a few things to help people give an answer useful to you: 1. What do you mean by "generalized tilt"? 2. Are you asking about a right circular cone? Does the angle at the vertex matter? 3. What form of answer (implicit, parametric...?) are you looking for? 4. If this is homework, what tools do you have available, an...
Read more

Color the edges and diagonals of a regular polygon

Image
Clash Royale CLAN TAG #URR8PPP up vote 5 down vote favorite 1 Here is the problem: For what $n$ is it possible to color the edges and diagonals of an $n$-side regular polygon with $dfracbinomn23$ colors, such that you use every color exactly three times and for every color the three segments (edges or diagonals) with that color form a triangle? Trivially $nequiv 0 text or 1 pmod3$. I can also prove that if the statement is true for $k$ than it is true for $3k$ as well. How to finish? Please help! Thanks combinatorial-geometry share | cite | improve this question edited Aug 6 at 21:57 alcana 118 4 asked Aug 6 at 21:15 Leo Gardner 356 11 Even assuming "polyhpn" in the title is a typo for polygon , it is unclear what you mean by "the edges and sides". Aren't the side of a regular polygon the same as the edges? A regular $n$-sided polygon has $n$ edges. – hardmath Aug 6 at 21:20 3 $n$ ne...
Read more

Relationship between determinant of matrix and determinant of adjoint?

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite We are studying adjoints in class, and I was curious if there is a relationship between the determinant of matrix A, and the determinant of the adjoint of matrix A? I assume there would be a relationship because finding the adjoint requires creating a cofactor matrix and then transposing it. linear-algebra determinant adjoint-operators share | cite | improve this question asked Nov 17 '17 at 17:32 CluelessCoder 32 5 For a square matrix $A$ of order $n$, we have $A(operatornameadj A)=(operatornameadj A)A=|A|I_n$ where $I_n$ is the identity matrix of order $n$. This should tell you about the determinant of the adjoint in terms of that of $A$ (use multiplicative property and other elementary properties of determinants). – Prasun Biswas Nov 17 '17 at 17:35 1 @CluelessCoder: see here: math.stackexchange.com/questions/516127/…...
Read more