Linearised polynomial

<h2 id="properties">Properties</h2>
<ul><li>The map x ↦ L(x) is a <a href="/facts/Linear_map/Q1PBKNVV">linear map</a> over any <a href="/facts/Field_(mathematics)/xAjAS4ko">field</a> containing Fq.</li>
<li>The <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> of roots of L is an Fq-<a href="/facts/Vector_space/M1pxMLx2">vector space</a> and is closed under the q-<a href="/facts/Frobenius_map/b9hkmc97">Frobenius map</a>.</li>
<li>Conversely, if U is any Fq-<a href="/facts/Linear_subspace/2wtKTgHL">linear subspace</a> of some finite field containing Fq, then the polynomial that vanishes exactly on U is a linearised polynomial.</li>
<li>The set of linearised polynomials over a given field is closed under addition and <a href="/facts/Function_composition/r5Pvnmth">composition</a> of polynomials.</li>
<li>If L is a nonzero linearised polynomial over 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
 with all of its roots lying in the field 
 
 
 
 
 F
 
 
 q
 
 s
 
 
 
 
 
 
 {\displaystyle F_{q^{s}}}
 
 an extension field of 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
, then each root of L has the same multiplicity, which is either 1, or a positive power of q.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li></ul>
<h2 id="symbolic-multiplication">Symbolic multiplication</h2>
In general, the product of two linearised polynomials will not be a linearized polynomial, but since the composition of two linearised polynomials results in a linearised polynomial, composition may be used as a replacement for multiplication and, for this reason, composition is often called symbolic multiplication in this setting. Notationally, if L1(x) and L2(x) are linearised polynomials we define 
 
 
 
 
 L
 
 1
 
 
 (
 x
 )
 ⊗
 
 L
 
 2
 
 
 (
 x
 )
 =
 
 L
 
 1
 
 
 (
 
 L
 
 2
 
 
 (
 x
 )
 )
 
 
 {\displaystyle L_{1}(x)\otimes L_{2}(x)=L_{1}(L_{2}(x))}
 
 when this point of view is being taken.

<h2 id="associated-polynomials">Associated polynomials</h2>
The polynomials L(x) and 
 
 
 
 l
 (
 x
 )
 =
 
 ∑
 
 i
 =
 0
 
 
 n
 
 
 
 a
 
 i
 
 
 
 x
 
 i
 
 
 
 
 {\displaystyle l(x)=\sum _{i=0}^{n}a_{i}x^{i}}
 
 are q-associates (note: the exponents "qi" of L(x) have been replaced by "i" in l(x)). More specifically, l(x) is called the conventional q-associate of L(x), and L(x) is the linearised q-associate of l(x).

<h2 id="q-polynomials-over-fq">q-polynomials over Fq</h2>
Linearised polynomials with coefficients in Fq have additional properties which make it possible to define symbolic division, symbolic reducibility and symbolic factorization. Two important examples of this type of linearised polynomial are the Frobenius automorphism 
 
 
 
 x
 ↦
 
 x
 
 q
 
 
 
 
 {\displaystyle x\mapsto x^{q}}
 
 and the trace function 
 
 
 
 Tr
 ⁡
 (
 x
 )
 =
 
 ∑
 
 i
 =
 0
 
 
 n
 −
 1
 
 
 
 x
 
 
 q
 
 i
 
 
 
 
 .
 
 
 {\textstyle \operatorname {Tr} (x)=\sum _{i=0}^{n-1}x^{q^{i}}.}

In this special case it can be shown that, as an <a href="/facts/Operation_(mathematics)/Yplv602Q">operation</a>, symbolic multiplication is <a href="/facts/Commutative_property/WaYxJLxd">commutative</a>, <a href="/facts/Associative/EQX8lV6r">associative</a> and <a href="/facts/Distributive_property/0LNBrVJl">distributes</a> over ordinary addition.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Also, in this special case, we can define the operation of symbolic division. If L(x) and L1(x) are linearised polynomials over Fq, we say that L1(x) symbolically divides L(x) if there exists a linearised polynomial L2(x) over Fq for which: 
 
 
 
 L
 (
 x
 )
 =
 
 L
 
 1
 
 
 (
 x
 )
 ⊗
 
 L
 
 2
 
 
 (
 x
 )
 .
 
 
 {\displaystyle L(x)=L_{1}(x)\otimes L_{2}(x).}
 
 
If L1(x) and L2(x) are linearised polynomials over Fq with conventional q-associates l1(x) and l2(x) respectively, then L1(x) symbolically divides L2(x) <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> l1(x) divides l2(x).<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> Furthermore, 
L1(x) divides L2(x) in the ordinary sense in this case.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>
A linearised polynomial L(x) over Fq of <a href="/facts/Degree_of_a_polynomial/Bf8vEIhf">degree</a> > 1 is symbolically irreducible over Fq if the only symbolic decompositions

L
        (
        x
        )
        =
        
          L
          
            1
          
        
        (
        x
        )
        ⊗
        
          L
          
            2
          
        
        (
        x
        )
        ,
      
    
    {\displaystyle L(x)=L_{1}(x)\otimes L_{2}(x),}

with Li over Fq are those for which one of the factors has degree 1. Note that a symbolically irreducible polynomial is always <a href="/facts/Reducible_polynomial/PTeuKm4W">reducible</a> in the ordinary sense since any linearised polynomial of degree > 1 has the nontrivial factor x. A linearised polynomial L(x) over Fq is symbolically irreducible if and only if its conventional q-associate l(x) is irreducible over Fq.
Every q-polynomial L(x) over Fq of degree > 1 has a symbolic factorization into symbolically irreducible polynomials over Fq and this factorization is essentially unique (up to rearranging factors and multiplying by nonzero elements of Fq.)
For example,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> consider the 2-polynomial L(x) = x16 + x8 + x2 + x over F2 and its conventional 2-associate l(x) = x4 + x3 + x + 1. The factorization into irreducibles of l(x) = (x2 + x + 1)(x + 1)2 in F2[x], gives the symbolic factorization

L
        (
        x
        )
        =
        (
        
          x
          
            4
          
        
        +
        
          x
          
            2
          
        
        +
        x
        )
        ⊗
        (
        
          x
          
            2
          
        
        +
        x
        )
        ⊗
        (
        
          x
          
            2
          
        
        +
        x
        )
        .
      
    
    {\displaystyle L(x)=(x^{4}+x^{2}+x)\otimes (x^{2}+x)\otimes (x^{2}+x).}

<h2 id="affine-polynomials">Affine polynomials</h2>
Let L be a linearised polynomial over 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
. A polynomial of the form 
 
 
 
 A
 (
 x
 )
 =
 L
 (
 x
 )
 −
 α
 
  for 
 
 α
 ∈
 
 F
 
 
 q
 
 n
 
 
 
 
 ,
 
 
 {\displaystyle A(x)=L(x)-\alpha {\text{ for }}\alpha \in F_{q^{n}},}
 
 is an affine polynomial over 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
.
Theorem: If A is a nonzero affine polynomial over 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
 with all of its roots lying in the field 
 
 
 
 
 F
 
 
 q
 
 s
 
 
 
 
 
 
 {\displaystyle F_{q^{s}}}
 
 an extension field of 
 
 
 
 
 F
 
 
 q
 
 n
 
 
 
 
 
 
 {\displaystyle F_{q^{n}}}
 
, then each root of A has the same multiplicity, which is either 1, or a positive power of q.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>

<h2 id="notes">Notes</h2>

<ul><li>Lidl, Rudolf; <a href="/facts/Harald_Niederreiter/MoayKG13">Niederreiter, Harald</a> (1997). <a href="https://archive.org/details/finitefields0000lidl_a8r3">Finite fields</a>. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-39231-4. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0866.11069">0866.11069</a>.</li>
<li>Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, Discrete Mathematics and its Applications, Boca Raton: CRC Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4398-7378-6</li></ul>

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

<ol>
<li id="fn:1">Lidl & Niederreiter 1997, pg.107 (first edition) - Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. <a href="https://archive.org/details/finitefields0000lidl_a8r3" target="_blank">https://archive.org/details/finitefields0000lidl_a8r3</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Mullen & Panario 2013, p. 23 (2.1.106) - Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, Discrete Mathematics and its Applications, Boca Raton: CRC Press, ISBN 978-1-4398-7378-6 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Lidl & Niederreiter 1997, pg. 115 (first edition) - Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. <a href="https://archive.org/details/finitefields0000lidl_a8r3" target="_blank">https://archive.org/details/finitefields0000lidl_a8r3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Lidl & Niederreiter 1997, pg. 115 (first edition) Corollary 3.60 - Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. <a href="https://archive.org/details/finitefields0000lidl_a8r3" target="_blank">https://archive.org/details/finitefields0000lidl_a8r3</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lidl & Niederreiter 1997, pg. 116 (first edition) Theorem 3.62 - Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. <a href="https://archive.org/details/finitefields0000lidl_a8r3" target="_blank">https://archive.org/details/finitefields0000lidl_a8r3</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Lidl & Niederreiter 1997, pg. 117 (first edition) Example 3.64 - Lidl, Rudolf; Niederreiter, Harald (1997). Finite fields. Encyclopedia of Mathematics and Its Applications. Vol. 20 (2nd ed.). Cambridge University Press. ISBN 0-521-39231-4. Zbl 0866.11069. <a href="https://archive.org/details/finitefields0000lidl_a8r3" target="_blank">https://archive.org/details/finitefields0000lidl_a8r3</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Mullen & Panario 2013, p. 23 (2.1.109) - Mullen, Gary L.; Panario, Daniel (2013), Handbook of Finite Fields, Discrete Mathematics and its Applications, Boca Raton: CRC Press, ISBN 978-1-4398-7378-6 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
</ol>

Linearised polynomial open-in-new

Linearised polynomial