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	~~Greetings~~. Let ~~me start by telling you~~ the ~~writer~~'~~s name~~ - ~~Phebe~~. ~~I used~~ to be ~~unemployed but now I am a librarian and the wage has been really~~ satisfying. ~~South Dakota~~ is ~~exactly where me and my husband live and my family members enjoys~~ it. ~~One~~ of the ~~extremely best things~~ in the ~~globe for him~~ is to ~~gather badges but he~~ is ~~having difficulties to find time~~ for it.<br><br>~~Review my page -~~ [~~http:~~//~~Bit~~.do/~~Lu9u Bit~~.do]		{{more footnotes\|date=March 2012}}
			In [[number theory]], '''Hilbert's irreducibility theorem''', conceived by [[David Hilbert]], states that every finite number of [[irreducible polynomial]]s in a finite number of variables and having [[rational number]] coefficients admit a common specialization of a proper subset of the variables to rational numbers such that all the polynomials remain irreducible. This theorem is a prominent theorem in number theory.

			== Formulation of the theorem ==
			'''Hilbert's irreducibility theorem.''' Let

			: <math>f_1(X_1,\ldots, X_r, Y_1,\ldots, Y_s), \ldots, f_n(X_1,\ldots, X_r, Y_1,\ldots, Y_s) \, </math>

			be irreducible polynomials in the ring

			:<math> \mathbb{Q}[X_1,\ldots, X_r, Y_1,\ldots, Y_s]. \, </math>

			Then there exists an ''r''-tuple of rational numbers (''a''<sub>1</sub>,...,''a''<sub>''r''</sub>) such that

			: <math>f_1(a_1,\ldots, a_r, Y_1,\ldots, Y_s), \ldots, f_n(a_1,\ldots, a_r, Y_1,\ldots, Y_s) \, </math>

			are irreducible in the ring

			:<math> \mathbb{Q}[Y_1,\ldots, Y_s]. \, </math>

			'''Remarks.'''
			* It follows from the theorem that there are infinitely many ''r''-tuples. In fact the set of all irreducible specialization, called Hilbert set, is large in many senses. For example, this set is [[Zariski topology\|Zariski dense]] in <math>\mathbb Q^r</math>

			* There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (''a''<sub>1</sub>,...,''a''<sub>''r''</sub>) to be integers.

			* There are many [[Hilbertian field]]s, i.e., fields satisfying Hilbert's irreducibility theorem. For example, [[global field]]s are Hilbertian.<ref name=L41>Lang (1997) p.41</ref>

			* The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take <math>n=r=s=1</math> in the definition. A recent result of Bary-Soroker shows that for a field ''K'' to be Hilbertian it suffices to consider the case of <math>n=r=s=1</math> and <math> f=f_1</math> [[absolutely irreducible]], that is, irreducible in the ring ''K''<sup>alg</sup>[''X'',''Y''], where ''K''<sup>alg</sup> is the algebraic closure of ''K''.

			== Applications ==
			Hilbert's irreducibility theorem has numerous applications in [[number theory]] and [[algebra]]. For example:

			* The [[inverse Galois problem]], Hilbert's original motivation. The theorem almost immediately implies that if a finite group ''G'' can be realized as the Galois group of a Galois extension ''N'' of
			:: <math>E=\mathbb{Q}(X_1,\ldots, X_r),</math>
			:then it can be specialized to a Galois extension ''N''<sub>0</sub> of the rational numbers with ''G'' as its Galois group.<ref name=L42>Lang (1997) p.42</ref> (To see this, choose a monic irreducible polynomial ''f''(''X''<sub>1</sub>,…,''X<sub>n</sub>'',''Y'') whose root generates ''N'' over ''E''. If ''f''(''a''<sub>1</sub>,…,''a<sub>n</sub>'',''Y'') is irreducible for some ''a<sub>i</sub>'', then a root of it will generate the asserted ''N''<sub>0</sub>.)

			* Construction of elliptic curves with large rank.<ref name=L42/>

			* Hilbert's irreducibility theorem is used as a step in the [[Andrew Wiles]] proof of [[Fermat's last theorem]].

			*If a polynomial <math>g(x) \in \mathbb{Z}[x]</math> is a perfect square for all large integer values of ''x'', then ''g(x)'' is the square of a polynomial in <math>\mathbb{Z}[x]</math>. This follows from Hilbert's irreducibility theorem with <math>n=r=s=1</math> and
			: <math>f_1(X, Y)\, = Y^2 - g(X)</math>.
			(More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.

			== Generalizations ==

			It has been reformulated and generalized extensively, by using the language of [[algebraic geometry]]. See [[thin set (Serre)]].

			== References ==
			{{reflist}}
			* {{cite book \| first=Serge \| last=Lang \| authorlink=Serge Lang \| title=Survey of Diophantine Geometry \| publisher=[[Springer-Verlag]] \| year=1997 \| isbn=3-540-61223-8 \| zbl=0869.11051 }}
			*J. P. Serre, ''Lectures on The Mordell-Weil Theorem'', Vieweg, 1989.
			*M. D. Fried and M. Jarden, ''Field Arithmetic'', Springer-Verlag, Berlin, 2005.
			*H. Völklein, ''Groups as Galois Groups'', Cambridge University Press, 1996.
			*G. Malle and B. H. Matzat, ''Inverse Galois Theory'', Springer, 1999.

			[[Category:Theorems in number theory]]
			[[Category:Polynomials]]
			[[Category:Theorems in algebra]]

en>Ferengi: sorting iw

2012-07-19T00:32:06Z

sorting iw

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Greetings. Let me start by telling you the writer's name - Phebe. I used to be unemployed but now I am a librarian and the wage has been really satisfying. South Dakota is exactly where me and my husband live and my family members enjoys it. One of the extremely best things in the globe for him is to gather badges but he is having difficulties to find time for it.<br><br>Review my page - [http://Bit.do/Lu9u Bit.do]

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170.74.55.81 at 20:49, 19 June 2014

87.6.95.89 at 13:10, 2 January 2014

en>Ferengi: sorting iw