Is it correct to say the field of complex numbers is contained in the field of quaternions?












2












$begingroup$


I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the set of quaternions. Please correct me if that is wrong.



Based on that assumption, I ask: is the field of complex numbers a subset (sub-field?) of the quaternions in the same sense that the real numbers are 'contained' in the complex numbers?



Is there a term for such a subordination of number fields?



I admit that I am reaching well beyond my realm of familiarity, and that my question may be nonsensical to those who are more knowledgeable in these matters.










share|cite|improve this question











$endgroup$












  • $begingroup$
    I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
    $endgroup$
    – DonAntonio
    3 hours ago












  • $begingroup$
    Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
    $endgroup$
    – reuns
    3 hours ago


















2












$begingroup$


I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the set of quaternions. Please correct me if that is wrong.



Based on that assumption, I ask: is the field of complex numbers a subset (sub-field?) of the quaternions in the same sense that the real numbers are 'contained' in the complex numbers?



Is there a term for such a subordination of number fields?



I admit that I am reaching well beyond my realm of familiarity, and that my question may be nonsensical to those who are more knowledgeable in these matters.










share|cite|improve this question











$endgroup$












  • $begingroup$
    I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
    $endgroup$
    – DonAntonio
    3 hours ago












  • $begingroup$
    Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
    $endgroup$
    – reuns
    3 hours ago
















2












2








2





$begingroup$


I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the set of quaternions. Please correct me if that is wrong.



Based on that assumption, I ask: is the field of complex numbers a subset (sub-field?) of the quaternions in the same sense that the real numbers are 'contained' in the complex numbers?



Is there a term for such a subordination of number fields?



I admit that I am reaching well beyond my realm of familiarity, and that my question may be nonsensical to those who are more knowledgeable in these matters.










share|cite|improve this question











$endgroup$




I believe it is correct to refer to the complex numbers and their 'native algebra' as a field. See for example Linear Algebra and Matrix Theory, by Evar Nering. I assume the same may be said for the set of quaternions. Please correct me if that is wrong.



Based on that assumption, I ask: is the field of complex numbers a subset (sub-field?) of the quaternions in the same sense that the real numbers are 'contained' in the complex numbers?



Is there a term for such a subordination of number fields?



I admit that I am reaching well beyond my realm of familiarity, and that my question may be nonsensical to those who are more knowledgeable in these matters.







abstract-algebra complex-numbers definition quaternions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 3 hours ago







Steven Hatton

















asked 3 hours ago









Steven HattonSteven Hatton

769315




769315












  • $begingroup$
    I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
    $endgroup$
    – DonAntonio
    3 hours ago












  • $begingroup$
    Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
    $endgroup$
    – reuns
    3 hours ago




















  • $begingroup$
    I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
    $endgroup$
    – DonAntonio
    3 hours ago












  • $begingroup$
    Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
    $endgroup$
    – reuns
    3 hours ago


















$begingroup$
I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
$endgroup$
– DonAntonio
3 hours ago






$begingroup$
I'm guessing you actually meant to ask whether the complex field can be embedded in the (Hamilton) quaternions $;Bbb H;$ . Now, this last is a divison ring, or a non-commutative "integral domain" (very cautious with this as integral domains' definition usually carries commutativity as conditon), sometimes also named "as skew-field, whereas $;Bbb C;$ is commutative...so you need an embedding into the center of $;Bbb H;$ : what is it?
$endgroup$
– DonAntonio
3 hours ago














$begingroup$
Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
$endgroup$
– reuns
3 hours ago






$begingroup$
Yes and no : $mathbb{C}$ is a field and $mathbb{H}$ is a skew-field, containing many copies of $mathbb{C}$ (but only one copy of $mathbb{R}$), $mathbb{H}$ is a non-commutative $mathbb{R}$-algebra with defining relations $j^2 = -1, zj = joverline{z}, z in mathbb{C}$ so it is not a $mathbb{C}$-algebra. In this sense it extends $mathbb{R}$, not $mathbb{C}$.
$endgroup$
– reuns
3 hours ago












3 Answers
3






active

oldest

votes


















3












$begingroup$

The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of complex numbers is isomorphic to a subfield of the complex field.



Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.






share|cite|improve this answer









$endgroup$





















    3












    $begingroup$

    The quaternion skew field ${Bbb H} = {Bbb C}times {Bbb C}$ has ${Bbb C}$ as a
    subfield by considering the ring monomorphism ${Bbb C}rightarrow {Bbb H}: zmapsto (z,0)$.






    share|cite|improve this answer









    $endgroup$





















      2












      $begingroup$

      The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.






      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%2f3075526%2fis-it-correct-to-say-the-field-of-complex-numbers-is-contained-in-the-field-of-q%23new-answer', 'question_page');
        }
        );

        Post as a guest















        Required, but never shown

























        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        3












        $begingroup$

        The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of complex numbers is isomorphic to a subfield of the complex field.



        Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.






        share|cite|improve this answer









        $endgroup$


















          3












          $begingroup$

          The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of complex numbers is isomorphic to a subfield of the complex field.



          Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.






          share|cite|improve this answer









          $endgroup$
















            3












            3








            3





            $begingroup$

            The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of complex numbers is isomorphic to a subfield of the complex field.



            Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.






            share|cite|improve this answer









            $endgroup$



            The field of complex numbers is isomorphic to a subfield of the division ring of the quaternions. Also, the field of complex numbers is isomorphic to a subfield of the complex field.



            Whether or not we have subsets here depends on the way the complex numbers and the quaternions are defined.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered 3 hours ago









            José Carlos SantosJosé Carlos Santos

            154k22123226




            154k22123226























                3












                $begingroup$

                The quaternion skew field ${Bbb H} = {Bbb C}times {Bbb C}$ has ${Bbb C}$ as a
                subfield by considering the ring monomorphism ${Bbb C}rightarrow {Bbb H}: zmapsto (z,0)$.






                share|cite|improve this answer









                $endgroup$


















                  3












                  $begingroup$

                  The quaternion skew field ${Bbb H} = {Bbb C}times {Bbb C}$ has ${Bbb C}$ as a
                  subfield by considering the ring monomorphism ${Bbb C}rightarrow {Bbb H}: zmapsto (z,0)$.






                  share|cite|improve this answer









                  $endgroup$
















                    3












                    3








                    3





                    $begingroup$

                    The quaternion skew field ${Bbb H} = {Bbb C}times {Bbb C}$ has ${Bbb C}$ as a
                    subfield by considering the ring monomorphism ${Bbb C}rightarrow {Bbb H}: zmapsto (z,0)$.






                    share|cite|improve this answer









                    $endgroup$



                    The quaternion skew field ${Bbb H} = {Bbb C}times {Bbb C}$ has ${Bbb C}$ as a
                    subfield by considering the ring monomorphism ${Bbb C}rightarrow {Bbb H}: zmapsto (z,0)$.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 2 hours ago









                    WuestenfuxWuestenfux

                    4,0501411




                    4,0501411























                        2












                        $begingroup$

                        The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.






                        share|cite|improve this answer









                        $endgroup$


















                          2












                          $begingroup$

                          The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.






                          share|cite|improve this answer









                          $endgroup$
















                            2












                            2








                            2





                            $begingroup$

                            The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.






                            share|cite|improve this answer









                            $endgroup$



                            The ring of quaternions(it is not a field) contains infinitely many subrings that are fields and are isomorphic to the field of complex numbers.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered 3 hours ago









                            Emilio NovatiEmilio Novati

                            51.7k43473




                            51.7k43473






























                                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%2f3075526%2fis-it-correct-to-say-the-field-of-complex-numbers-is-contained-in-the-field-of-q%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

                                Magento 2 controller redirect on button click in phtml file

                                Polycentropodidae