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Gk where the lower right submatrix taken modulo I yields Sk (f , g), where all entries in the lower left submatrix are in I, and the upper left submatrix is an upper m−j triangular matrix with fn ’s on the diagonal. Thus we obtain resk (f, g) = fn · resk (f , g), and since fnm−j ∈ I the assertion follows also in this case. 17) Proposition. Let R be a principal ideal domain, let p ∈ R be a prime, and let : R → R/ p =: F denote the natural map; note that F is a field. Moreover, let 0 = f, g ∈ R[X] such that p | gcd(lc(f ), lc(g)) ∈ R, and let h := gcd(f, g) ∈ R[X].

E. we have v + v , w = v, w + v , w and vλ, w = λ v, w as well as w, v = v, w , for all v, v , w ∈ V and λ ∈ K. In particular, we have v, wλ = λ v, w . For U ≤ V let U ⊥ := {v ∈ V ; v, u = 0 for all u ∈ U } ≤ V . In particular, rad( ·, · ) := V ⊥ is called the radical of ·, · . The form ·, · is called non-degenerate if rad( ·, · ) = {0}. A vector 0 = v ∈ V is called isotropic, if v, v = 0; the form ·, · is called anisotropic if there are no isotropic vectors. If K = R the form ·, · just is a symmetric bilinear form.

N}, where {e1 , . . , en } ⊆ Cn is the standard C-basis. Then for n v = [v1 , . . , vn ] ∈ Cn we have q(v) = v, v = i=1 |vi |2 ∈ R≥0 , the standard n 2 quadratic form, hence ||v|| := q(v) = i=1 |vi | =∈ R≥0 is the 2-norm. Let A = [aij ] ∈ Cn×n , and let ai := [ai1 , . . , ain ] ∈ Cn denote its rows, for i ∈ {1, . . , n}. Then we have | det(A)| ≤ n i=1 ||ai || = n i=1 n j=1 |aij |2 1 2 . Proof. We may assume that det(A) = 0, and let B := {a1 , . . , an } ⊆ Cn , which hence is a C-basis. Let B = {a1 , .

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