@misc{FIELD_9.MIZ,
      AUTHOR = {Schwarzweller, Christoph and
        Rowin\'{n}ska-Schwarzweller, Agnieszka},
      TITLE = {Quadratic Extensions},
      ADDRESS1 = {Institute of Informatics, University of Gdansk, Poland},
      ADDRESS2 = {Sopot, Poland},
      SECTION1 = {Preliminaries},
      SECTION2 = {More on Polynomials},
      SECTION3 = {Quadratic Polynomials},
      SECTION4 = {Quadratic Polynomials over $Z/2$},
      SECTION5 = {Fields with non-squares},
      SECTION6 = {Splittingfields for quadratic Polynomials},
      SECTION7 = {Quadratic Extensions},
      DAY = {30},
      MONTH = {November},
      YEAR = {2021},
      SUMMARY = {In this article we further develop field theory in
                 Mizar \cite{Mizar-State-2015,BancerekJAR:2018,GrabKornSchwarz:2016}:
                 We deal with quadratic polynomials and quadratic extensions.
                 First we introduce quadratic polynomials, their disciminants and
                 proof the midnight formula. Then we show that - in case the                  discriminant
                 of $p$ being non-square - adjoining a root of $p$'s discriminant
                 results in a splitting field of $p$. Finally we prove that these are
                 the only field extensions of degree 2, e.g. that an extension $E$ of                   $F$
                 is quadratic if and only if there is a non square Element $a \in $F$
                 such that $E$ and $F(\sqrt{a})$ are isomorphic over F.}}

@BOOK{HL99,
      AUTHOR = {H. L\"{u}neburg},
      TITLE = {Gruppen, Ringe, K\"{o}rper: Die grundlegenden Strukturen der Algebra},
      PUBLISHER = {Oldenbourg Verlag},
      YEAR = {1990}}

@BOOK{Lan02,
      AUTHOR = {Serge Lang},
      TITLE = {Algebra, {R}evised {T}hird {E}dition},
      PUBLISHER = {Springer Verlag},
      YEAR = {2002}}

@BOOK{Jac85,
      AUTHOR = {Nathan Jacobson},
      TITLE = {Basic Algebra I},
      PUBLISHER = {Dover Books on Mathematics},
      YEAR = {1985}}
