By N. H. Abel, E. Galois

Zu der hinterlassenen Abllamllullg VOll Abel, S. 57-81. 1 Die Definition der Ordnung eines algebraischen Ausdrucks, wie sie auf Seite sixty seven gegeben ist, ist incorrcct und nach der auf S. 10 angefiihrten zu berichtigen. Die Ordnung eines algebraischen Ausdrucks ist additionally nicht gleich der Anzahl der in ihm ausser den bekannten Grossen auftretenden Wurzelgrossen, sondern vielmehr, wenn guy sich des Symbols V-Wie ublich zur Bezeichnung der Wurzelgrossen bedient, gleich der grossten von denjenigen Zahlen, welche angeben, wie viele solcher Wurzelzeichen sich in dem gegebenen algebraischen Ausdruck uber einander erstrecken. Dabei wird vorausgesetzt, dass, wenn ein Wurzelzeichen einen Index hat, welcher eine zusammengesetzte Zahl ist, dasselbe nach der Formel 1Jtn m -V-= VFso weit umgeformt werde, bis siimtliche Wurzelzeiehen Primzahl exponenten tragen, und dass sich keines dieser Wurzelzeichen durch Ausfuhrung der durch dasselbe angedeuteten Operation beseitigen Hisst. Kommen in einem algebraischen Ausdruck mehrere solcher auf einander oder auf algebrai. che Ausdrucke niederer Ordnung nicht reducierbarer Wurzelgrossen vor, in denen jene, die grosste Anzahl der iiber einander sich erstreekenden 'Wurzelzeichen angebenden Zahlen einander gleich sind, so giebt die Anzahl derselben den Grad des algebraischen Ausdrucks an. - Ist In die Ordnung des algebraischen Ausdrucks und bezeichnet guy die einzelnen Wurzelgrossen in der Reihenfolge, wie sie numerisch berechnet werden ter mussen, um den Wert der Wurzelgrosse m Ordnung zu erhalten, mit ""m-l . . . .

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Additional resources for Abhandlungen ueber die algebraische Aufloesung der Gleichungen

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Let us see that the projection πr : Fr := pP8 (p−1 N is birational. R (r)) In fact, Fr is of dimension 3 and irreducible and the intersection of a threecodimensional linear subspace and two quadrics in P8 . 1 (2), (ii). Thus for a general point P in N , Fr ∩ Lr , P consists of Lr and a single point (namely, the point of intersection of the two lines which are the residual intersections of each of the two quadrics defining Fr with Lr , P , the other component being Lr itself). Thus πr is generically one-to-one whence birational.

Let us look at V (2, 2) for example. Here we get three concentric regular hexagons (one of them is degenerate and consists of the origin alone). The weights are thus 2π1 + 2π2 , 3π2 , −2π1 + 4π2 , −3π1 + 3π2 , −4π1 + 2π2 , −3π1 , −2π1 − 2π2 , −3π2 , 2π1 − 4π2 , 3π1 − 3π2 , 4π1 − 2π2 , 3π1 (these are the ones on the outer hexagon, read counterclockwise, and have multiplicity 1), π1 + π2 , −π1 + 2π2 , −2π1 + π2 , −π1 − π2 , π1 − 2π2 , 2π1 − π2 (these lie on the middle hexagon and have multiplicity 2), and finally there is 0 with multiplicity 3 corresponding to the origin.

For f ∈ V (0, 4) and g ∈ V (2, 2) one defines a bilinear SL3 C-equivariant map α : V (0, 4) × V (2, 2) → Sym2 C3 ⊗ Sym3 (C3 )∨ , (f, g) → α(f, g), as the contraction sij11 ij22 i3 := f i1 i2 i4 i5 gii56 ji31 det−1 j2 i4 i6 , followed by the symmetrization map. One checks that Sym2 C3 ⊗ Sym3 (C3 )∨ decomposes as V (2, 3) ⊕ V (1, 2) ⊕ V (0, 1), but Sym4 V (0, 4) does not contain these as subrepresentations (use Magma), so α(f, Ψ (f )) = 0 for all f ∈ V (0, 4). But the explicit forms of Ψ and α show that ker α(·, Ψ (h20 )) = Ch20 , whence, by upper-semicontinuity, the dimension of the kernel of α(·, Ψ (f )) is one for all f in a dense open subset of V (0, 4), and the rational map Ψ : PV (0, 4) Ψ PV (0, 4) ⊂ PV (2, 2) has the rational inverse Ψ (f ) → ker α(·, Ψ (f )).

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Abhandlungen ueber die algebraische Aufloesung der Gleichungen by N. H. Abel, E. Galois

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