{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T12:08:02Z","timestamp":1777464482233,"version":"3.51.4"},"reference-count":10,"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 first part of a four-article series containing a Mizar [3], [1], [2] 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                    [9], [4], [6].\n                  <\/jats:p>\n                  <jats:p>\n                    In this 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<\/jats:italic>\n                    :\n                    <jats:italic>F<\/jats:italic>\n                    \u2192\n                    <jats:italic>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 the 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<\/jats:italic>\n                    \u2192\n                    <jats:italic>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 \u2229 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 \\ 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<\/jats:italic>\n                    <jats:sup>\u2032<\/jats:sup>\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                    ] = \u2205and \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                    i\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 \u2286 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<\/jats:italic>\n                    \u2192\n                    <jats:italic>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<\/jats:italic>\n                    \u2229\n                    <jats:italic>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-2019-0010","type":"journal-article","created":{"date-parts":[[2019,7,22]],"date-time":"2019-07-22T05:30:34Z","timestamp":1563773434000},"page":"93-100","source":"Crossref","is-referenced-by-count":4,"title":["On Roots of Polynomials over\n                    <i>F<\/i>\n                    [\n                    <i>X<\/i>\n                    ]\/ \u3008\n                    <i>p<\/i>\n                    \u3009"],"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":"2026042801383774883_j_forma-2019-0010_ref_001_w2aab3b7ab1b6b1ab1ab1Aa","doi-asserted-by":"crossref","unstructured":"[1] Grzegorz Bancerek, Czes\u0142aw Byli\u0144ski, Adam Grabowski, Artur Korni\u0142owicz, Roman Matuszewski, Adam Naumowicz, Karol P\u0105k, and Josef Urban. 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