Write the algebraic closure of $F_p$ as union of finite fields












4












$begingroup$


In Field theory by Steven Roman Chapter 9 Exercise 20, if we write the algebraic closure of finite field $F_q$ as $Gamma(q)$ and $a_n$ be any strictly increasing infinite sequence of positive integers, the exercise wants to prove that $Gamma(q)=bigcup_{n=0}^{infty}GF(q^{a_n})$.



However, if $a_n$ is an arbitrary sequence, we are even unable to prove $bigcup_{n=0}^{infty}GF(q^{a_n})$ is a field. I wonder whether the exercise has omitted some condition since the equality doesn't hold under current conditions offered.



Hope for answers!






abstract-algebra field-theory finite-fields







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    any sequence of positive integers such that any $k$ divides some $a_n$
    $endgroup$
    – reuns
    Dec 7 '18 at 20:29










  • $begingroup$
    @reuns That condition is both necessary and sufficient, right?
    $endgroup$
    – Wembley Inter
    Dec 7 '18 at 20:47






  • 2




    $begingroup$
    No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:11






  • 5




    $begingroup$
    My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:18








  • 5




    $begingroup$
    The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
    $endgroup$
    – Martin Sleziak
    Dec 8 '18 at 7:04
















4












$begingroup$


In Field theory by Steven Roman Chapter 9 Exercise 20, if we write the algebraic closure of finite field $F_q$ as $Gamma(q)$ and $a_n$ be any strictly increasing infinite sequence of positive integers, the exercise wants to prove that $Gamma(q)=bigcup_{n=0}^{infty}GF(q^{a_n})$.



However, if $a_n$ is an arbitrary sequence, we are even unable to prove $bigcup_{n=0}^{infty}GF(q^{a_n})$ is a field. I wonder whether the exercise has omitted some condition since the equality doesn't hold under current conditions offered.



Hope for answers!






abstract-algebra field-theory finite-fields







share|cite|improve this question











$endgroup$








  • 5




    $begingroup$
    any sequence of positive integers such that any $k$ divides some $a_n$
    $endgroup$
    – reuns
    Dec 7 '18 at 20:29










  • $begingroup$
    @reuns That condition is both necessary and sufficient, right?
    $endgroup$
    – Wembley Inter
    Dec 7 '18 at 20:47






  • 2




    $begingroup$
    No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:11






  • 5




    $begingroup$
    My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:18








  • 5




    $begingroup$
    The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
    $endgroup$
    – Martin Sleziak
    Dec 8 '18 at 7:04














4












4








4





$begingroup$


In Field theory by Steven Roman Chapter 9 Exercise 20, if we write the algebraic closure of finite field $F_q$ as $Gamma(q)$ and $a_n$ be any strictly increasing infinite sequence of positive integers, the exercise wants to prove that $Gamma(q)=bigcup_{n=0}^{infty}GF(q^{a_n})$.



However, if $a_n$ is an arbitrary sequence, we are even unable to prove $bigcup_{n=0}^{infty}GF(q^{a_n})$ is a field. I wonder whether the exercise has omitted some condition since the equality doesn't hold under current conditions offered.



Hope for answers!






abstract-algebra field-theory finite-fields







share|cite|improve this question











$endgroup$




In Field theory by Steven Roman Chapter 9 Exercise 20, if we write the algebraic closure of finite field $F_q$ as $Gamma(q)$ and $a_n$ be any strictly increasing infinite sequence of positive integers, the exercise wants to prove that $Gamma(q)=bigcup_{n=0}^{infty}GF(q^{a_n})$.



However, if $a_n$ is an arbitrary sequence, we are even unable to prove $bigcup_{n=0}^{infty}GF(q^{a_n})$ is a field. I wonder whether the exercise has omitted some condition since the equality doesn't hold under current conditions offered.



Hope for answers!






abstract-algebra field-theory finite-fields




abstract-algebra field-theory finite-fields






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 8 '18 at 8:41









Brahadeesh

6,22742361




6,22742361










asked Dec 7 '18 at 20:26









Wembley InterWembley Inter

1215




1215








  • 5




    $begingroup$
    any sequence of positive integers such that any $k$ divides some $a_n$
    $endgroup$
    – reuns
    Dec 7 '18 at 20:29










  • $begingroup$
    @reuns That condition is both necessary and sufficient, right?
    $endgroup$
    – Wembley Inter
    Dec 7 '18 at 20:47






  • 2




    $begingroup$
    No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:11






  • 5




    $begingroup$
    My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:18








  • 5




    $begingroup$
    The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
    $endgroup$
    – Martin Sleziak
    Dec 8 '18 at 7:04














  • 5




    $begingroup$
    any sequence of positive integers such that any $k$ divides some $a_n$
    $endgroup$
    – reuns
    Dec 7 '18 at 20:29










  • $begingroup$
    @reuns That condition is both necessary and sufficient, right?
    $endgroup$
    – Wembley Inter
    Dec 7 '18 at 20:47






  • 2




    $begingroup$
    No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:11






  • 5




    $begingroup$
    My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
    $endgroup$
    – Lubin
    Dec 8 '18 at 5:18








  • 5




    $begingroup$
    The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
    $endgroup$
    – Martin Sleziak
    Dec 8 '18 at 7:04








5




5




$begingroup$
any sequence of positive integers such that any $k$ divides some $a_n$
$endgroup$
– reuns
Dec 7 '18 at 20:29




$begingroup$
any sequence of positive integers such that any $k$ divides some $a_n$
$endgroup$
– reuns
Dec 7 '18 at 20:29












$begingroup$
@reuns That condition is both necessary and sufficient, right?
$endgroup$
– Wembley Inter
Dec 7 '18 at 20:47




$begingroup$
@reuns That condition is both necessary and sufficient, right?
$endgroup$
– Wembley Inter
Dec 7 '18 at 20:47




2




2




$begingroup$
No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
$endgroup$
– Lubin
Dec 8 '18 at 5:11




$begingroup$
No, @usr0192, for instance $Bbb F_4notsubsetBbb F_8$.
$endgroup$
– Lubin
Dec 8 '18 at 5:11




5




5




$begingroup$
My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
$endgroup$
– Lubin
Dec 8 '18 at 5:18






$begingroup$
My earlier comment seems to have gotten lost: this author has been unacceptably sloppy in asking you to prove something that just isn’t so. The suggestion of @reuns will work, as will the specification of ${a_n}$ to be the sequence of factorials: $a_n=n!$. I think the best way of looking at this problem is to consider $Gamma(q)$ to be the direct limit of the finite fields $Bbb F_{q^n}$, with respect to the directed set of the positive integers, ordered by divisibility.
$endgroup$
– Lubin
Dec 8 '18 at 5:18






5




5




$begingroup$
The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 7:04




$begingroup$
The same question on MathOverflow: Write the algebra closure of $F_p$ as union of finite fields. This answer has a very reasonable advice about cross-posting - and you can also looks at other discussions on cross-posting.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 7:04










1 Answer
1






active

oldest

votes


















4












$begingroup$

Since the question is now closed on MathOF (and rather belongs on here), here's an answer in the form of an exercise ((a),(b) below). Yes, the exercise in Roman's book is false and it is quite hardly reassuring that such an exercise appears as published in a GTM book.



First, to be rigorous, we should work within one given algebraically closed field $K$ with characteristic $p$ (where $p$ is the unique prime divisor of $q$), so that $mathrm{GF}(q^k)$ is a well-defined subfield of $K$, and so that the union makes sense as a subset of $K$.



(a) As suggested by @reuns (and copied by the OP to the MO post without reference), for a sequence $a=(a_n)$, the union $L_a=bigcup_nmathrm{GF}(q^{a_n})$ is an algebraic closed subfield of $K$ iff for every $kge 1$ there exists $n$ such that $k$ divides $a_n$;



(b) this union $L_a$ is a subfield of $K$ iff for any $m,m'$, there exists $n$ such that $mathrm{lcm}(a_m,a_{m'})$ divides $a_n$.



(c) for every subfield $L$ of $K$ algebraic over $mathrm{GF}(q)$, there exists a sequence $a=(a_n)$, such that $a_n$ divides $a_{n+1}$ for every $n$, such that $L=L_a$.



For instance, if $a_n=2^n$ for all $n$, then the union is a field, but not algebraically closed. If $a_n$ is the $n$-th prime, the union is not a field.






share|cite|improve this answer











$endgroup$













    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',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    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%2f3030335%2fwrite-the-algebraic-closure-of-f-p-as-union-of-finite-fields%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    4












    $begingroup$

    Since the question is now closed on MathOF (and rather belongs on here), here's an answer in the form of an exercise ((a),(b) below). Yes, the exercise in Roman's book is false and it is quite hardly reassuring that such an exercise appears as published in a GTM book.



    First, to be rigorous, we should work within one given algebraically closed field $K$ with characteristic $p$ (where $p$ is the unique prime divisor of $q$), so that $mathrm{GF}(q^k)$ is a well-defined subfield of $K$, and so that the union makes sense as a subset of $K$.



    (a) As suggested by @reuns (and copied by the OP to the MO post without reference), for a sequence $a=(a_n)$, the union $L_a=bigcup_nmathrm{GF}(q^{a_n})$ is an algebraic closed subfield of $K$ iff for every $kge 1$ there exists $n$ such that $k$ divides $a_n$;



    (b) this union $L_a$ is a subfield of $K$ iff for any $m,m'$, there exists $n$ such that $mathrm{lcm}(a_m,a_{m'})$ divides $a_n$.



    (c) for every subfield $L$ of $K$ algebraic over $mathrm{GF}(q)$, there exists a sequence $a=(a_n)$, such that $a_n$ divides $a_{n+1}$ for every $n$, such that $L=L_a$.



    For instance, if $a_n=2^n$ for all $n$, then the union is a field, but not algebraically closed. If $a_n$ is the $n$-th prime, the union is not a field.






    share|cite|improve this answer











    $endgroup$


















      4












      $begingroup$

      Since the question is now closed on MathOF (and rather belongs on here), here's an answer in the form of an exercise ((a),(b) below). Yes, the exercise in Roman's book is false and it is quite hardly reassuring that such an exercise appears as published in a GTM book.



      First, to be rigorous, we should work within one given algebraically closed field $K$ with characteristic $p$ (where $p$ is the unique prime divisor of $q$), so that $mathrm{GF}(q^k)$ is a well-defined subfield of $K$, and so that the union makes sense as a subset of $K$.



      (a) As suggested by @reuns (and copied by the OP to the MO post without reference), for a sequence $a=(a_n)$, the union $L_a=bigcup_nmathrm{GF}(q^{a_n})$ is an algebraic closed subfield of $K$ iff for every $kge 1$ there exists $n$ such that $k$ divides $a_n$;



      (b) this union $L_a$ is a subfield of $K$ iff for any $m,m'$, there exists $n$ such that $mathrm{lcm}(a_m,a_{m'})$ divides $a_n$.



      (c) for every subfield $L$ of $K$ algebraic over $mathrm{GF}(q)$, there exists a sequence $a=(a_n)$, such that $a_n$ divides $a_{n+1}$ for every $n$, such that $L=L_a$.



      For instance, if $a_n=2^n$ for all $n$, then the union is a field, but not algebraically closed. If $a_n$ is the $n$-th prime, the union is not a field.






      share|cite|improve this answer











      $endgroup$
















        4












        4








        4





        $begingroup$

        Since the question is now closed on MathOF (and rather belongs on here), here's an answer in the form of an exercise ((a),(b) below). Yes, the exercise in Roman's book is false and it is quite hardly reassuring that such an exercise appears as published in a GTM book.



        First, to be rigorous, we should work within one given algebraically closed field $K$ with characteristic $p$ (where $p$ is the unique prime divisor of $q$), so that $mathrm{GF}(q^k)$ is a well-defined subfield of $K$, and so that the union makes sense as a subset of $K$.



        (a) As suggested by @reuns (and copied by the OP to the MO post without reference), for a sequence $a=(a_n)$, the union $L_a=bigcup_nmathrm{GF}(q^{a_n})$ is an algebraic closed subfield of $K$ iff for every $kge 1$ there exists $n$ such that $k$ divides $a_n$;



        (b) this union $L_a$ is a subfield of $K$ iff for any $m,m'$, there exists $n$ such that $mathrm{lcm}(a_m,a_{m'})$ divides $a_n$.



        (c) for every subfield $L$ of $K$ algebraic over $mathrm{GF}(q)$, there exists a sequence $a=(a_n)$, such that $a_n$ divides $a_{n+1}$ for every $n$, such that $L=L_a$.



        For instance, if $a_n=2^n$ for all $n$, then the union is a field, but not algebraically closed. If $a_n$ is the $n$-th prime, the union is not a field.






        share|cite|improve this answer











        $endgroup$



        Since the question is now closed on MathOF (and rather belongs on here), here's an answer in the form of an exercise ((a),(b) below). Yes, the exercise in Roman's book is false and it is quite hardly reassuring that such an exercise appears as published in a GTM book.



        First, to be rigorous, we should work within one given algebraically closed field $K$ with characteristic $p$ (where $p$ is the unique prime divisor of $q$), so that $mathrm{GF}(q^k)$ is a well-defined subfield of $K$, and so that the union makes sense as a subset of $K$.



        (a) As suggested by @reuns (and copied by the OP to the MO post without reference), for a sequence $a=(a_n)$, the union $L_a=bigcup_nmathrm{GF}(q^{a_n})$ is an algebraic closed subfield of $K$ iff for every $kge 1$ there exists $n$ such that $k$ divides $a_n$;



        (b) this union $L_a$ is a subfield of $K$ iff for any $m,m'$, there exists $n$ such that $mathrm{lcm}(a_m,a_{m'})$ divides $a_n$.



        (c) for every subfield $L$ of $K$ algebraic over $mathrm{GF}(q)$, there exists a sequence $a=(a_n)$, such that $a_n$ divides $a_{n+1}$ for every $n$, such that $L=L_a$.



        For instance, if $a_n=2^n$ for all $n$, then the union is a field, but not algebraically closed. If $a_n$ is the $n$-th prime, the union is not a field.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 9 '18 at 11:15


























        community wiki





        2 revs
        YCor































            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3030335%2fwrite-the-algebraic-closure-of-f-p-as-union-of-finite-fields%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more