Hilbert's irreducibility theorem

<h2 id="formulation-of-the-theorem">Formulation of the theorem</h2>
Hilbert's irreducibility theorem. Let

f
          
            1
          
        
        (
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            r
          
        
        ,
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        )
        ,
        …
        ,
        
          f
          
            n
          
        
        (
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            r
          
        
        ,
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        )
      
    
    {\displaystyle 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})}

be irreducible polynomials in the ring

Q
        
        (
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            r
          
        
        )
        [
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        ]
        .
      
    
    {\displaystyle \mathbb {Q} (X_{1},\ldots ,X_{r})[Y_{1},\ldots ,Y_{s}].}

Then there exists an r-tuple of rational numbers (a1, ..., ar) such that

f
          
            1
          
        
        (
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            r
          
        
        ,
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        )
        ,
        …
        ,
        
          f
          
            n
          
        
        (
        
          a
          
            1
          
        
        ,
        …
        ,
        
          a
          
            r
          
        
        ,
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        )
      
    
    {\displaystyle 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})}

are irreducible in the ring

Q
        
        [
        
          Y
          
            1
          
        
        ,
        …
        ,
        
          Y
          
            s
          
        
        ]
        .
      
    
    {\displaystyle \mathbb {Q} [Y_{1},\ldots ,Y_{s}].}

Remarks.

<ul><li>It follows from the theorem that there are infinitely many r-tuples. In fact the set of all irreducible specializations, called Hilbert set, is large in many senses. For example, this set is <a href="/facts/Zariski_topology/uolkKBpM">Zariski dense</a> in 
 
 
 
 
 
 Q
 
 
 r
 
 
 .
 
 
 {\displaystyle \mathbb {Q} ^{r}.}
 
</li>
<li>There are always (infinitely many) integer specializations, i.e., the assertion of the theorem holds even if we demand (a1, ..., ar) to be integers.</li>
<li>There are many <a href="/facts/Hilbertian_field/aVBYjqNe">Hilbertian fields</a>, i.e., fields satisfying Hilbert's irreducibility theorem. For example, <a href="/facts/Number_field/e0K0TXSh">number fields</a> are Hilbertian.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a></li>
<li>The irreducible specialization property stated in the theorem is the most general. There are many reductions, e.g., it suffices to take 
 
 
 
 n
 =
 r
 =
 s
 =
 1
 
 
 {\displaystyle n=r=s=1}
 
 in the definition. A result of Bary-Soroker shows that for a field K to be Hilbertian it suffices to consider the case of 
 
 
 
 n
 =
 r
 =
 s
 =
 1
 
 
 {\displaystyle n=r=s=1}
 
 and 
 
 
 
 f
 =
 
 f
 
 1
 
 
 
 
 {\displaystyle f=f_{1}}
 
 <a href="/facts/Absolutely_irreducible/VrNqEn5c">absolutely irreducible</a>, that is, irreducible in the ring Kalg[X,Y], where Kalg is the algebraic closure of K.</li></ul>
<h2 id="applications">Applications</h2>
Hilbert's irreducibility theorem has numerous applications in <a href="/facts/Number_theory/zhoa7zrc">number theory</a> and <a href="/facts/Algebra/nMdqczjo">algebra</a>. For example:

<ul><li>The <a href="/facts/Inverse_Galois_problem/phfoWPLl">inverse Galois problem</a>, 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</li></ul>

E
        =
        
          Q
        
        (
        
          X
          
            1
          
        
        ,
        …
        ,
        
          X
          
            r
          
        
        )
        ,
      
    
    {\displaystyle E=\mathbb {Q} (X_{1},\ldots ,X_{r}),}

then it can be specialized to a Galois extension N0 of the rational numbers with G as its Galois group.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> (To see this, choose a monic irreducible polynomial f(X1, ..., Xn, Y) whose root generates N over E. If f(a1, ..., an, Y) is irreducible for some ai, then a root of it will generate the asserted N0.)
<ul><li>Construction of elliptic curves with large rank.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a></li></ul>
<ul><li>Hilbert's irreducibility theorem is used as a step in the <a href="/facts/Andrew_Wiles/bdc3k3pQ">Andrew Wiles</a> proof of <a href="/facts/Fermat%2527s_Last_Theorem/fHRc2t8z">Fermat's Last Theorem</a>.</li></ul>
<ul><li>If a polynomial 
 
 
 
 g
 (
 x
 )
 ∈
 
 Z
 
 [
 x
 ]
 
 
 {\displaystyle g(x)\in \mathbb {Z} [x]}
 
 is a perfect square for all large integer values of x, then g(x) is the square of a polynomial in 
 
 
 
 
 Z
 
 [
 x
 ]
 .
 
 
 {\displaystyle \mathbb {Z} [x].}
 
 This follows from Hilbert's irreducibility theorem with 
 
 
 
 n
 =
 r
 =
 s
 =
 1
 
 
 {\displaystyle n=r=s=1}
 
 and</li></ul>

f
          
            1
          
        
        (
        X
        ,
        Y
        )
        =
        
          Y
          
            2
          
        
        −
        g
        (
        X
        )
        .
      
    
    {\displaystyle f_{1}(X,Y)=Y^{2}-g(X).}

(More elementary proofs exist.) The same result is true when "square" is replaced by "cube", "fourth power", etc.
<h2 id="generalizations">Generalizations</h2>
It has been reformulated and generalized extensively, by using the language of <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>. See <a href="/facts/Thin_set_(Serre)/aVBYjqNe">thin set (Serre)</a>.

<ul><li>D. Hilbert, "Uber die Irreducibilitat ganzer rationaler Functionen mit ganzzahligen Coefficienten", J. reine angew. Math. 110 (1892) 104–129.</li></ul>

<ul><li><a href="/facts/Serge_Lang/I7MW2rUu">Lang, Serge</a> (1997). Survey of Diophantine Geometry. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-61223-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0869.11051">0869.11051</a>.</li>
<li>J. P. Serre, Lectures on The Mordell-Weil Theorem, Vieweg, 1989.</li>
<li>M. D. Fried and M. Jarden, Field Arithmetic, Springer-Verlag, Berlin, 2005.</li>
<li>H. Völklein, Groups as Galois Groups, Cambridge University Press, 1996.</li>
<li>G. Malle and B. H. Matzat, Inverse Galois Theory, Springer, 1999.</li></ul>

<h2 id="references">References</h2>

Hilbert's irreducibility theorem open-in-new

Hilbert's irreducibility theorem