{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T15:07:33Z","timestamp":1777475253463,"version":"3.51.4"},"reference-count":5,"publisher":"Walter de Gruyter GmbH","issue":"2","license":[{"start":{"date-parts":[[2019,7,1]],"date-time":"2019-07-01T00:00:00Z","timestamp":1561939200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-sa\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"<jats:title>Summary<\/jats:title>\n                  <jats:p>\n                    This is the second part of a four-article series containing a Mizar [2], [1] formalization of Kronecker\u2019s construction about roots of polynomials in field extensions, i.e. that for every field\n                    <jats:italic>F<\/jats:italic>\n                    and every polynomial\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\\F<\/jats:italic>\n                    there exists a field extension\n                    <jats:italic>E<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    such that\n                    <jats:italic>p<\/jats:italic>\n                    has a root over\n                    <jats:italic>E<\/jats:italic>\n                    . The formalization follows Kronecker\u2019s classical proof using\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    as the desired field extension\n                    <jats:italic>E<\/jats:italic>\n                    [5], [3], [4].\n                  <\/jats:p>\n                  <jats:p>\n                    In the first part we show that an irreducible polynomial\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\\F<\/jats:italic>\n                    has a root over\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    . Note, however, that this statement cannot be true in a rigid formal sense: We do not have\n                    <jats:italic>F<\/jats:italic>\n                    \u2286 [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/ &lt; p &gt;<\/jats:italic>\n                    as sets, so\n                    <jats:italic>F<\/jats:italic>\n                    is not a subfield of\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    , and hence formally\n                    <jats:italic>p<\/jats:italic>\n                    is not even a polynomial over\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/ &lt; p &gt;<\/jats:italic>\n                    . Consequently, we translate\n                    <jats:italic>p<\/jats:italic>\n                    along the canonical monomorphism\n                    <jats:italic>\u03d5 : F \u2192 F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    and show that the translated polynomial\n                    <jats:italic>\u03d5<\/jats:italic>\n                    (\n                    <jats:italic>p<\/jats:italic>\n                    ) has a root over\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    .\n                  <\/jats:p>\n                  <jats:p>\n                    Because\n                    <jats:italic>F<\/jats:italic>\n                    is not a subfield of\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    we construct in this second part the field (\n                    <jats:italic>E \\ \u03d5F<\/jats:italic>\n                    )\u222a\n                    <jats:italic>F<\/jats:italic>\n                    for a given monomorphism\n                    <jats:italic>\u03d5 : F \u2192 E<\/jats:italic>\n                    and show that this field both is isomorphic to\n                    <jats:italic>F<\/jats:italic>\n                    and includes\n                    <jats:italic>F<\/jats:italic>\n                    as a subfield. In the literature this part of the proof usually consists of saying that \u201cone can identify\n                    <jats:italic>F<\/jats:italic>\n                    with its image\n                    <jats:italic>\u03d5F<\/jats:italic>\n                    in\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    and therefore consider\n                    <jats:italic>F<\/jats:italic>\n                    as a subfield of\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    \u201d. Interestingly, to do so we need to assume that\n                    <jats:italic>F<\/jats:italic>\n                    \u2229\n                    <jats:italic>E<\/jats:italic>\n                    = \u2205, in particular Kronecker\u2019s construction can be formalized for fields\n                    <jats:italic>F<\/jats:italic>\n                    with\n                    <jats:italic>F<\/jats:italic>\n                    \u2229\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ] = \u2205.\n                  <\/jats:p>\n                  <jats:p>\n                    Surprisingly, as we show in the third part, this condition is not automatically true for arbitray fields\n                    <jats:italic>F<\/jats:italic>\n                    : With the exception of \ud835\udd51\n                    <jats:sub>2<\/jats:sub>\n                    we construct for every field\n                    <jats:italic>F<\/jats:italic>\n                    an isomorphic copy\n                    <jats:italic>F\u2032<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    with\n                    <jats:italic>F\u2032<\/jats:italic>\n                    \u2229\n                    <jats:italic>F\u2032<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\u2260<\/jats:italic>\n                    \u2205. We also prove that for Mizar\u2019s representations of \ud835\udd51\n                    <jats:sub>n<\/jats:sub>\n                    , \ud835\udd48 and \ud835\udd49 we have \ud835\udd51\n                    <jats:sub>n<\/jats:sub>\n                    \u2229 \ud835\udd51\n                    <jats:sub>n<\/jats:sub>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ] = \u2205, \ud835\udd48 \u2229 \ud835\udd48 [\n                    <jats:italic>X<\/jats:italic>\n                    ] = \u2205 and \ud835\udd49 \u2229 \ud835\udd49 [\n                    <jats:italic>X<\/jats:italic>\n                    ] = \u2205, respectively.\n                  <\/jats:p>\n                  <jats:p>\n                    In the fourth part we finally define field extensions:\n                    <jats:italic>E<\/jats:italic>\n                    is a field extension of\n                    <jats:italic>F<\/jats:italic>\n                    iff\n                    <jats:italic>F<\/jats:italic>\n                    is a subfield of\n                    <jats:italic>E<\/jats:italic>\n                    . Note, that in this case we have\n                    <jats:italic>F<\/jats:italic>\n                    \u2286\n                    <jats:italic>E<\/jats:italic>\n                    as sets, and thus a polynomial\n                    <jats:italic>p<\/jats:italic>\n                    over\n                    <jats:italic>F<\/jats:italic>\n                    is also a polynomial over\n                    <jats:italic>E<\/jats:italic>\n                    . We then apply the construction of the second part to\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    with the canonical monomorphism\n                    <jats:italic>\u03d5 : F \u2192 F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\/&lt;p&gt;<\/jats:italic>\n                    . Together with the first part this gives - for fields\n                    <jats:italic>F<\/jats:italic>\n                    with\n                    <jats:italic>F \u2229 F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ] = \u2205 - a field extension\n                    <jats:italic>E<\/jats:italic>\n                    of\n                    <jats:italic>F<\/jats:italic>\n                    in which\n                    <jats:italic>p<\/jats:italic>\n                    \u2208\n                    <jats:italic>F<\/jats:italic>\n                    [\n                    <jats:italic>X<\/jats:italic>\n                    ]\n                    <jats:italic>\\F<\/jats:italic>\n                    has a root.\n                  <\/jats:p>","DOI":"10.2478\/forma-2019-0014","type":"journal-article","created":{"date-parts":[[2019,7,22]],"date-time":"2019-07-22T05:30:45Z","timestamp":1563773445000},"page":"133-137","source":"Crossref","is-referenced-by-count":2,"title":["On Monomorphisms and Subfields"],"prefix":"10.2478","volume":"27","author":[{"given":"Christoph","family":"Schwarzweller","sequence":"first","affiliation":[{"name":"Institute of Informatics , University of Gda\u0144sk , Poland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2019,7,20]]},"reference":[{"key":"2026042801413510038_j_forma-2019-0014_ref_001_w2aab3b7b4b1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, and Karol P\u0105k. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9\u201332, 2018. doi:10.1007\/s10817-017-9440-6.10.1007\/s10817-017-9440-6604425130069070","DOI":"10.1007\/s10817-017-9440-6"},{"key":"2026042801413510038_j_forma-2019-0014_ref_002_w2aab3b7b4b1b6b1ab1ab2Aa","doi-asserted-by":"crossref","unstructured":"[2] Adam Grabowski, Artur Korni\u0142owicz, and Adam Naumowicz. Four decades of Mizar. Journal of Automated Reasoning, 55(3):191\u2013198, 2015. doi:10.1007\/s10817-015-9345-1.10.1007\/s10817-015-9345-1","DOI":"10.1007\/s10817-015-9345-1"},{"key":"2026042801413510038_j_forma-2019-0014_ref_003_w2aab3b7b4b1b6b1ab1ab3Aa","unstructured":"[3] Nathan Jacobson. Basic Algebra I. Dover Books on Mathematics, 1985."},{"key":"2026042801413510038_j_forma-2019-0014_ref_004_w2aab3b7b4b1b6b1ab1ab4Aa","doi-asserted-by":"crossref","unstructured":"[4] Heinz L\u00fcneburg. Gruppen, Ringe, K\u00f6rper: Die grundlegenden Strukturen der Algebra. Oldenbourg Verlag, 1999.10.1524\/9783486599022","DOI":"10.1524\/9783486599022"},{"key":"2026042801413510038_j_forma-2019-0014_ref_005_w2aab3b7b4b1b6b1ab1ab5Aa","unstructured":"[5] Knut Radbruch. Algebra I. Lecture Notes, University of Kaiserslautern, Germany, 1991."}],"container-title":["Formalized Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/content.sciendo.com\/view\/journals\/forma\/27\/2\/article-p133.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/reference-global.com\/pdf\/10.2478\/forma-2019-0014","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T13:52:08Z","timestamp":1777384328000},"score":1,"resource":{"primary":{"URL":"https:\/\/reference-global.com\/article\/10.2478\/forma-2019-0014"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,7,1]]},"references-count":5,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,7,20]]},"published-print":{"date-parts":[[2019,7,1]]}},"alternative-id":["10.2478\/forma-2019-0014"],"URL":"https:\/\/doi.org\/10.2478\/forma-2019-0014","relation":{},"ISSN":["1898-9934","1426-2630"],"issn-type":[{"value":"1898-9934","type":"electronic"},{"value":"1426-2630","type":"print"}],"subject":[],"published":{"date-parts":[[2019,7,1]]}}}