\providecommand{\bysame}{\leavevmode ---\ }
\providecommand{\og}{``}
\providecommand{\fg}{''}
\providecommand{\smfandname}{\&}
\providecommand{\smfedsname}{\'eds.}
\providecommand{\smfedname}{\'ed.}
\providecommand{\smfmastersthesisname}{M\'emoire}
\providecommand{\smfphdthesisname}{Th\`ese}
\begin{thebibliography}{CTKS87}

\bibitem[Art82]{artin}
{\scshape M.~Artin} -- {\og Brauer-{S}everi varieties\fg}, in \emph{Brauer
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  in Math., vol. 917, Springer, Berlin, 1982, p.~194--210.

\bibitem[AT68]{at}
{\scshape E.~Artin {\normalfont \smfandname} J.~Tate} -- \emph{Class field
  theory}, W. A. Benjamin, Inc., New York-Amsterdam, 1968.

\bibitem[BEL91]{bel}
{\scshape A.~Bertram, L.~Ein {\normalfont \smfandname} R.~Lazarsfeld} -- {\og
  Vanishing theorems, a theorem of {S}everi, and the equations defining
  projective varieties\fg}, \emph{J. Amer. Math. Soc.} \textbf{4} (1991),
  no.~3, p.~587--602.

\bibitem[Bri]{bright}
{\scshape M.~Bright} -- {\og Efficient evaluation of the {B}rauer-{M}anin
  obstruction\fg}, \emph{Math. Proc. Cambridge Philos. Soc.}, to appear.

\bibitem[Bro94]{brown}
{\scshape K.~S. Brown} -- \emph{Cohomology of groups}, Graduate Texts in
  Mathematics, vol.~87, Springer-Verlag, New York, 1994, Corrected reprint of
  the 1982 original.

\bibitem[BSD04]{bsd}
{\scshape M.~Bright {\normalfont \smfandname} P.~Swinnerton-Dyer} -- {\og
  Computing the {B}rauer-{M}anin obstructions\fg}, \emph{Math. Proc. Cambridge
  Philos. Soc.} \textbf{137} (2004), no.~1, p.~1--16.

\bibitem[CF67]{cf}
{\scshape J.~W.~S. Cassels {\normalfont \smfandname} A.~Fr{\"o}hlich}
  (\smfedsname) -- \emph{Algebraic number theory}, London, Academic Press,
  1967.

\bibitem[CG66]{cg}
{\scshape J.~W.~S. Cassels {\normalfont \smfandname} M.~J.~T. Guy} -- {\og On
  the {H}asse principle for cubic surfaces\fg}, \emph{Mathematika} \textbf{13}
  (1966), p.~111--120.

\bibitem[CLO97]{clo}
{\scshape D.~Cox, J.~Little {\normalfont \smfandname} D.~O'Shea} --
  \emph{Ideals, varieties, and algorithms}, second \smfedname, Undergraduate
  Texts in Mathematics, Springer-Verlag, New York, 1997.

\bibitem[CTKS87]{ctks}
{\scshape J.-L. Colliot-Th{\'e}l{\`e}ne, D.~Kanevsky {\normalfont \smfandname}
  J.-J. Sansuc} -- {\og Arithm\'etique des surfaces cubiques diagonales\fg}, in
  \emph{Diophantine approximation and transcendence theory (Bonn, 1985)},
  Lecture Notes in Math., vol. 1290, Springer, Berlin, 1987, p.~1--108.

\bibitem[CTS80]{cts}
{\scshape J.-L. Colliot-Th{\'e}l{\`e}ne {\normalfont \smfandname} J.-J. Sansuc}
  -- {\og La descente sur les vari\'et\'es rationnelles\fg}, in
  \emph{Journ\'ees de G\'eometrie Alg\'ebrique d'Angers, Juillet 1979/Algebraic
  Geometry, Angers, 1979}, Sijthoff \& Noordhoff, Alphen aan den Rijn, 1980,
  p.~223--237.

\bibitem[CTS87]{cts2}
{\scshape J.-L. Colliot-Th{\'e}l{\`e}ne {\normalfont \smfandname} J.-J. Sansuc}
  -- {\og La descente sur les vari\'et\'es rationnelles. {II}\fg}, \emph{Duke
  Math. J.} \textbf{54} (1987), no.~2, p.~375--492.

\bibitem[dJ05]{dejong}
{\scshape A.~J. de~Jong} -- {\og A result of {G}abber\fg}, 2005, preprint.

\bibitem[Dra83]{draxl}
{\scshape P.~K. Draxl} -- \emph{Skew fields}, London Mathematical Society
  Lecture Note Series, vol.~81, Cambridge University Press, Cambridge, 1983.

\bibitem[Gro60]{egaI}
{\scshape A.~Grothendieck} -- {\og \'{E}l\'ements de g\'eom\'etrie
  alg\'ebrique. {I}. {L}e langage des sch\'emas\fg}, \emph{Inst. Hautes
  \'Etudes Sci. Publ. Math.} (1960), no.~4.

\bibitem[Gro61]{egaII}
\bysame , {\og \'{E}l\'ements de g\'eom\'etrie alg\'ebrique. {II}. \'{E}tude
  globale \'el\'ementaire de quelques classes de morphismes\fg}, \emph{Inst.
  Hautes \'Etudes Sci. Publ. Math.} (1961), no.~8.

\bibitem[Gro68]{gb}
{\scshape A.~Grothendieck} -- {\og Le groupe de {B}rauer. {I}--{III}\fg}, in
  \emph{Dix Expos\'es sur la Cohomologie des Sch\'emas}, North-Holland,
  Amsterdam, 1968, p.~46--188.

\bibitem[Gro71]{sga1}
{\scshape A.~Grothendieck} -- \emph{Rev\^etements \'etales et groupe
  fondamental}, Springer-Verlag, Berlin, 1971, S\'eminaire de G\'eom\'etrie
  Alg\'ebrique du Bois Marie 1960--1961 (SGA 1), Dirig\'e par A. Grothendieck.
  Augment\'e de deux expos\'es de M. Raynaud, Lecture Notes in Mathematics,
  Vol. 224.

\bibitem[Has23a]{hasse2}
{\scshape H.~Hasse} -- {\og {\"U}ber die {{\"A}}quivalenz quadratischer
  {F}ormen im {K}\"orper der rationalen {Z}ahlen\fg}, \emph{J. Reine Angew.
  Math.} \textbf{152} (1923), p.~205--224.

\bibitem[Has23b]{hasse1}
\bysame , {\og {\"U}ber die {D}arstellbarkeit von {Z}ahlen durch quadratische
  {F}ormen im {K}\"orper der rationalen {Z}ahlen\fg}, \emph{J. Reine Angew.
  Math.} \textbf{152} (1923), p.~129--148.

\bibitem[Has24]{hasse3}
\bysame , {\og Darstellbarkeit von {Z}ahlen durch quadratische {F}ormen in
  einem beliebigen algebraischen {Z}ahlk\"orper\fg}, \emph{J. Reine Angew.
  Math.} \textbf{153} (1924), p.~113--130.

\bibitem[Hen08]{hensel08}
{\scshape K.~Hensel} -- \emph{Theorie der algebraischen {Z}ahlen}, B. G.
  Teubner, Leipzig, 1908.

\bibitem[Hen13]{hensel13}
\bysame , \emph{Zahlentheorie}, G. J. G\"oschen'sche Verlagshandlung, Berlin,
  1913.

\bibitem[HRR91]{hrr}
{\scshape J.~Heintz, T.~Recio {\normalfont \smfandname} M.-F. Roy} -- {\og
  Algorithms in real algebraic geometry and applications to computational
  geometry\fg}, in \emph{Discrete and computational geometry (New Brunswick,
  NJ, 1989/1990)}, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol.~6,
  Amer. Math. Soc., Providence, RI, 1991, p.~137--163.

\bibitem[IP99]{ip}
{\scshape V.~A. Iskovskikh {\normalfont \smfandname} Y.~G. Prokhorov} -- {\og
  Fano varieties\fg}, in \emph{Algebraic geometry, V}, Encyclopaedia Math.
  Sci., vol.~47, Springer, Berlin, 1999, p.~1--247.

\bibitem[KPS01]{kps}
{\scshape T.~Krick, L.~M. Pardo {\normalfont \smfandname} M.~Sombra} -- {\og
  Sharp estimates for the arithmetic {N}ullstellensatz\fg}, \emph{Duke Math.
  J.} \textbf{109} (2001), no.~3, p.~521--598.

\bibitem[KR00]{kr}
{\scshape M.~Kreuzer {\normalfont \smfandname} L.~Robbiano} --
  \emph{Computational commutative algebra. 1}, Springer-Verlag, Berlin, 2000.

\bibitem[KT04]{kt}
{\scshape A.~Kresch {\normalfont \smfandname} Y.~Tschinkel} -- {\og On the
  arithmetic of del {P}ezzo surfaces of degree 2\fg}, \emph{Proc. London Math.
  Soc. (3)} \textbf{89} (2004), no.~3, p.~545--569.

\bibitem[Lem03]{lemmermeyer}
{\scshape F.~Lemmermeyer} -- {\og A note on {P}\'epin's counter examples to the
  {H}asse principle for curves of genus 1\fg}, 2003, preprint.

\bibitem[Lin40]{lind}
{\scshape C.-E. Lind} -- {\og Untersuchungen \"uber die rationalen {P}unkte der
  ebenen kubischen {K}urven vom {G}eschlecht {E}ins\fg}, 1940, Thesis,
  University of Uppsala.

\bibitem[Log04]{logan}
{\scshape A.~Logan} -- {\og \texttt{Brauer}, {M}agma package for determining
  the {B}rauer--{M}anin obstruction of a del {P}ezzo surface of degree 4\fg},
  2004.

\bibitem[Man71]{manin}
{\scshape Y.~I. Manin} -- {\og Le groupe de {B}rauer-{G}rothendieck en
  g\'eom\'etrie diophantienne\fg}, in \emph{Actes du Congr\`es International
  des Math\'ematiciens (Nice, 1970), Tome 1}, Gauthier-Villars, Paris, 1971,
  p.~401--411.

\bibitem[Man74]{maninbook}
{\scshape Y.~I. Manin} -- \emph{Cubic forms: algebra, geometry, arithmetic},
  North-Holland Publishing Co., Amsterdam, 1974.

\bibitem[Min90]{minkowski}
{\scshape H.~Minkowski} -- {\og {\"U}ber die {B}edingungen, unter welchen zwei
  quadratische {F}ormen mit rationalen {K}oeffizienten ineinander rational
  transformiert werden k\"onnen\fg}, \emph{J. Reine Angew. Math.} \textbf{106}
  (1890), p.~5--26.

\bibitem[P{\'e}p80]{pepin}
{\scshape T.~P{\'e}pin} -- {\og Nouveaux th\'eor\`emes sur l'\'equation
  ind\'etermin\'ee $ax^4+by^4=z^2$\fg}, \emph{C. R. Acad. Sci Paris}
  \textbf{91} (1880), no.~2, p.~100--101.

\bibitem[Rei42]{reichardt}
{\scshape H.~Reichardt} -- {\og Einige im {K}leinen \"uberall l\"osbare, im
  {G}rossen unl\"osbare diophantische {G}leichungen\fg}, \emph{J. Reine Angew.
  Math.} \textbf{184} (1942), p.~12--18.

\bibitem[Ser68]{serre}
{\scshape J.-P. Serre} -- \emph{Corps locaux}, Hermann, Paris, 1968, Deuxi\`eme
  \'edition, Publications de l'Universit\'e de Nancago, No. VIII.

\bibitem[Siu93]{siu}
{\scshape Y.~T. Siu} -- {\og An effective {M}atsusaka big theorem\fg},
  \emph{Ann. Inst. Fourier (Grenoble)} \textbf{43} (1993), no.~5,
  p.~1387--1405.

\bibitem[Sko99]{skoro}
{\scshape A.~N. Skorobogatov} -- {\og Beyond the {M}anin obstruction\fg},
  \emph{Invent. Math.} \textbf{135} (1999), no.~2, p.~399--424.

\bibitem[Sou17]{sousley}
{\scshape C.~P. Sousley} -- {\og Invariants and covariants of the {C}remona
  cubic surface\fg}, \emph{Amer. J. Math.} \textbf{39} (1917), p.~135--146.

\bibitem[Wit04]{wittenberg}
{\scshape O.~Wittenberg} -- {\og Transcendental {B}rauer-{M}anin obstruction on
  a pencil of elliptic curves\fg}, in \emph{Arithmetic of higher-dimensional
  algebraic varieties (Palo Alto, CA, 2002)}, Progr. Math., vol. 226,
  Birkh\"auser Boston, Boston, MA, 2004, p.~259--267.

\end{thebibliography}
