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author={Schneeberger, William Alan},
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title={Universal quadratic forms and the 290-theorem},
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url={http://alpha.math.uga.edu/%7Epete/quadraticforms.pdf},
urldate={2023-10-22},
note={Lecture notes.}
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title={Introduction to quadratic forms},
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title={Disquisitiones arithmeticae},
author={Gauss, Carl Friedrich},
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title={Disquisitiones arithmeticae},
author={Gauss, Carl Friedrich},
year={1801},
publisher={Springer},
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title={Essai sur la theorie des nombres},
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title={Finite-dimensional vector spaces},
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title={Advanced linear algebra},
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title={A history of mathematics: An introduction},
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title={Rational quadratic forms},
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title={Introduction to the arithmetic theory of quadratic forms},
author={Raskin, Sam},
url={https://gauss.math.yale.edu/~sr2532/forms.pdf},
urldate={2023-10-30},
note={Lecture notes.}
}
@book{serre2012course,
title={A course in arithmetic},
author={Serre, Jean-Pierre},
volume={7},
year={2012},
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title={Les Nombres p-adiques},
author={Amice, Yvette},
year={1975},
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title={Annotations to Serre's Course in Arithmetic},
author={Weismann, Martin and Fassler, Jordan and Maisch, Filix},
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url={https://people.ucsc.edu/~weissman/Math222A/SerreAnn.pdf},
urldate={2023-10-30},
note={Lecture notes.}
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title={Theorie der quadratischen Formen in beliebigen K{\"o}rpern.},
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title={Basic quadratic forms},
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title={Number Theory: An approach through history from Hammurapi to Legendre},
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title={The elements of Euclid},
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title={Euclid's Elements},
note={From the edition of Johan Ludvig Heiberg, translated by Richard Fitzpatrick},
author={Euclid},
url={https://farside.ph.utexas.edu/books/Euclid/Elements.pdf},
urldate={2023-10-30}
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title={Modular elliptic curves and Fermat's last theorem},
author={Wiles, Andrew John},
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title={Zahlentheorie},
author={Hensel, Kurt},
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@phdthesis{hasse1922uber,
author = {Hasse, Helmut},
school = {Universit{\"a}t G{\"o}ttingen},
title = {Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen},
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author = {Slattery, Tom},
title = {The Representation of Integers by Quadratic Forms and the Conway-Schneeberger Fifteen Theorem},
year = {2020},
institution= {University of Warwick},
url = {https://www.academia.edu/43627690/The_Representation_of_Integers_by_Quadratic_Forms_and_the_Conway_Schneeberger_Fifteen_Theorem},
urldate = {2023-11-03}
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url={https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2014},
author={International Mathematical Union},
title={Fields Medals 2014 Awardees},
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title={Advanced modern algebra},
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title={Die theorie der algebraischen Zahlk{\"o}rper},
author={Hilbert, David},
booktitle={Jahresbericht der Deutschen Mathematiker-Vereinigung},
pages={175--546},
year={1895},
publisher={Springer}
}
@article{hermite1850extraits,
title={Extraits de lettres de M. Ch. Hermite {\`a} M. Jacobi sur diff{\'e}rents objects de la th{\'e}orie des nombres.},
author={Hermite, Charles},
journal={Journal f{\"u}r die reine und angewandte Mathematik},
volume={1850},
number={40},
year={1850},
publisher={De Gruyter Berlin},
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urldate={2023-11-03}
}
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title={An introduction to the geometry of numbers},
author={Cassels, J.W.S.},
year={1961},
publisher={Springer}
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title={Minkowski reduction of integral matrices},
author={Donaldson, John L.},
journal={Mathematics of Computation},
volume={33},
number={145},
pages={201--216},
year={1979}
}
@article{helfrich1985algorithms,
title={Algorithms to construct Minkowski reduced and Hermite reduced lattice bases},
author={Helfrich, Bettina},
journal={Theoretical Computer Science},
volume={41},
pages={125--139},
year={1985},
publisher={Elsevier}
}
@article{minkowski1885untersuchungen,
title={Untersuchungen {\"u}ber quadratische Formen: I. Bestimmung der Anzahl verschiedener Formen, welche ein gegebenes Genus enth{\"a}lt},
author={Minkowski, Hermann},
year={1885},
journal={Mathematische Abhandlungen}
}
@incollection{schwermer2007reduction,
author = {Schwermer, Joachim},
booktitle = {The shaping of arithmetic after C.F. Gauss's Disquisitiones arithmeticae},
publisher = {Springer},
title = {Reduction theory of quadratic forms: toward \emph{R\"aumliche Anschauung} in Minkowski's early work},
year = {2007},
pages = {483--502},
editor = {Goldstein, Catherine and Schappacher, Norbert and Schwermer, Joachim}
}
@incollection{goldstein2007book,
author = {Goldstein, Catherine and Schappacher, Norbert},
booktitle = {The shaping of arithmetic after C.F. Gauss's Disquisitiones arithmeticae},
publisher = {Springer},
title = {A book in search of a discipline (1801--1860)},
year = {2007},
pages = {3--66},
editor = {Goldstein, Catherine and Schappacher, Norbert and Schwermer, Joachim}
}
@article{tammela1979reduction,
title={Reduction theory of positive quadratic forms},
author={Tammela, P.P.},
journal={Journal of Soviet Mathematics},
volume={11},
pages={197--277},
year={1979},
publisher={Springer}
}
@article{van1956reduktionstheorie,
title={Die Reduktionstheorie der positiven quadratischen Formen},
author={van der Waerden, BL},
journal={Acta Mathematica},
volume={96},
number={1},
pages={265--309},
year={1956},
publisher={Kluwer Academic Publishers Dordrecht}
}
@article{mahler1938minkowski,
title={On Minkowski's theory of reduction of positive definite quadratic forms},
author={Mahler, Kurt},
journal={The Quarterly Journal of Mathematics},
number={1},
pages={259--262},
year={1938},
publisher={Oxford University Press}
}
@article{barnes1979minkowski,
title={Minkowski's fundamental inequality for reduced positive quadratic forms (II)},
author={Barnes, E.S.},
journal={Journal of the Australian Mathematical Society},
volume={27},
number={1},
pages={1--6},
year={1979},
publisher={Cambridge University Press}
}
@article{van1969dasminimum,
title={Das Minimum von \(D/f_{11}f_{12}\cdots f_{55}\) f{\"u}r reduzierte positive quin\"are quadratische Formen},
author={van der Waerden, B.L.},
journal={Aequationes Mathematicae},
volume={2},
pages={233--247},
year={1969}
}
@article{omeara1955quadratic,
author = {O'Meara, O.T.},
journal = {American Journal of Mathematics},
number = {1},
title = {Quadratic forms over local fields},
volume = {77},
year = {1955},
page = {87--116}
}
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title={Equivalence of quadratic forms},
author={Siegel, Carl L.},
journal={American Journal of Mathematics},
volume={63},
number={3},
pages={658--680},
year={1941}
}
@book{dudley1978elementary,
title={Elementary number theory},
author={Dudley, Underwood},
year={1978},
publisher={W.H. Freeman and Company}
}
@online{clark2011representation,
title={Representation of integers by quadratic forms},
author={Clark, Pete L.},
url={http://alpha.math.uga.edu/~pete/4400theend2011.pdf},
urldate={2023-11-15}
}
@online{sutherland2013introduction10,
title={Lecture 10: Hilbert symbols},
author={Sutherland, Andrew},
url={https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/196f7a627ae6d75ab009bbef96e7c8c9_MIT18_782F13_lec10.pdf},
urldate={2023-11-15},
note={Lecture notes for MIT course 18.782, ``Introduction to Arithmetic Geometry.''}
}
@online{sutherland2013introduction04,
title={Lecture 4: The ring of $p$-adic integers},
author={Sutherland, Andrew},
url = {https://ocw.mit.edu/courses/18-782-introduction-to-arithmetic-geometry-fall-2013/resources/mit18_782f13_lec4/},
urldate={2023-11-15},
note={Lecture notes for MIT course 18.782, ``Introduction to Arithmetic Geometry.''}
}
@online{poonen2009introduction,
title={Introduction to arithmetic geometry},
author={Poonen, Bjorn},
url={https://math.mit.edu/~poonen/782/782notes.pdf},
urldate={2023-11-15},
note={Lecture notes for MIT course 18.783, ``Topics in Algebraic Number Theory.''}
}
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title={Principles of mathematical analysis},
author={Rudin, Walter},
year={1976},
publisher={McGraw-Hill New York}
}
@article{dickson1927integers,
title={Integers represented by positive ternary quadratic forms},
author={Dickson, Leonard Eugene},
year={1927},
journal={Bulletin of the American Mathematical Society}
}
@article{kim2004recent,
title={Recent developments on universal forms},
author={Kim, Myung-Hwan},
journal={Contemporary Mathematics},
volume={344},
pages={215--228},
year={2004},
publisher={Providence, RI: American Mathematical Society}
}
@article{bhargavageneralization,
title={The fifteen theorem and generalizations},
author={Manjul Bhargava},
note={preprint}
}
@article{kneser1954theorie,
title={Zur Theorie der Kristallgitter},
author={Kneser, Martin},
journal={Mathematische Annalen},
volume={127},
number={1},
pages={105--106},
year={1954},
publisher={Springer}
}
@article{ramanujan1916onthe,
title={On the expression of a number in the form $ax^2 + by^2 + cz^2 + du^2$},
author={Ramanujan, Srivinasa},
year={1960},
journal={Proceedings of the Cambridge Philosophical Society},
volume = {19},
pages = {11--21},
}
@article{willerding1948determination,
title={Determination of all classes of positive quaternary quadratic forms which represent all (positive) integers},
author={Willerding, Margaret F},
year={1948},
journal={Bulletin of the American Mathematical Society},
volume = {54},
pages = {334-337},
}
@inproceedings{schulze1991algorithm,
title={An algorithm for computing genera of ternary and quaternary quadratic forms},
author={Schulze-Pillot, Rainer},
booktitle={Proceedings of the 1991 international symposium on Symbolic and algebraic computation},
pages={134--143},
year={1991}
}