Consider the field K = Q ( X 1 , … , X p ), where X is your nonzero solution. We want to construct a ( Q -linear) map f : K → Q with some f ( X i ) ≠ 0 since the entries of A lie in Q , we'll then have A f ( X ) = 0 with f ( X ) ∈ Q p nonzero. Construct a tower of fields Q = K 0 ⊂ ⋯ ⊂ K r = K with each K i / K i − 1 either algebraic or isomorphic to K i − 1 ( t ) for some purely transcendental element t. In the former case, use the nondegeneracy of the trace form. In the latter case, use the fact that the map f : K i − 1 ( t ) → K i − 1 with f | K i − 1 = id and f ( t ) = α for any fixed α ∈ K i − 1 is well-defined.

Question

Consider the field   K  =      Q    (      X    1    ,  …  ,      X    p    ), where   X is your nonzero solution. We want to construct a (      Q  -linear) map   f  :  K  →      Q   with some   f  (      X    i    )  ≠  0 since the entries of   A lie in       Q  , we&#039;ll then have   A  f  (  X  )  =  0 with   f  (  X  )  ∈            Q        p   nonzero. Construct a tower of fields       Q    =      K    0    ⊂  ⋯  ⊂      K    r    =  K with each       K    i        /        K          i      −      1       either algebraic or isomorphic to       K          i      −      1        (  t  ) for some purely transcendental element   t. In the former case, use the nondegeneracy of the trace form. In the latter case, use the fact that the map   f  :      K          i      −      1        (  t  )  →      K          i      −      1       with   f  |      K          i      −      1        =  id and   f  (  t  )  =  α for any fixed   α  ∈      K          i      −      1       is well-defined.

Warkallent8 · Accepted Answer

Consider the field K=Q(X1,…,Xp), where X is your nonzero solution. We want to construct a (Q-linear) map f:K->Q with some f(Xi)≠0 since the entries of A lie in Q, we'll then have Af(X)=0 with f(X) in Qp nonzero.

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