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Algebraic Geometry

By Jean-Pierre Serre

ISBN-10: 0201093847

ISBN-13: 9780201093841

This vintage publication comprises an creation to platforms of l-adic representations, a subject of significant significance in quantity idea and algebraic geometry, as mirrored via the remarkable contemporary advancements at the Taniyama-Weil conjecture and Fermat's final Theorem. The preliminary chapters are dedicated to the Abelian case (complex multiplication), the place one reveals a pleasant correspondence among the l-adic representations and the linear representations of a few algebraic teams (now known as Taniyama groups). The final bankruptcy handles the case of elliptic curves with out advanced multiplication, the most results of that's that a dead ringer for the Galois workforce (in the corresponding l-adic illustration) is "large."

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Sd ] K[s1 ] respecting the G-action. Assume now that K(V ) is the splitting field over K(V )G of a specialisation F (t, X), v = (t1 , . . , tr ) ∈ (K(V )G )r . For a suitable w ∈ K[V ] \ 0, we have that t1 , . . , the ring of elements of the form a/we for a ∈ K[V ] and e ∈ N), and also that F (v, X) ∈ K[V ]w [X]. Moreover, we can — for each σ ∈ G \ 1 – pick a root ξ ∈ K[V ]w of F (t, X) with σξ = ξ and require 1/(σξ − ξ) ∈ K[V ]w . Let w be the image of w in K[s1 , . . , sd ], and pick the qi ’s as above to ensure that w maps to a non-zero element w ∈ K[s1 ].

We then have homomorphisms K[V ]w → K[s1 , . . , sd ]w K[s1 ]w , all respecting the G-action. 14]), find θ ∈ M such that θ = (σθ)σ∈G is a normal basis for M/L and w (θ) = 0. Thus, we have K[V ]w → K[s1 , . . , sd ]w K[s1 ]w → M, with the last map defined as follows: If s1 = (sσ )σ∈G with σsτ = sστ , we map sσ to σθ. This gives us a K-algebra homomorphism K[V ]w → M respecting the Gaction. Letting a = (a1 , . . , ar ) be the images of t in M , we see that a1 , . . , ar ∈ L and that F (a, X) splits completely in M [X].

Groups of Degree 4 4 3 Let f (X) = X + a3 X + a2 X 2 + a1 X + a0 ∈ K[X] be irreducible, where (once again) K is a field of characteristic = 2. In this case, the discriminant d(f ) of f (X) cannot alone determine the structure of the Galois group of f (X) over K. , the symmetry group of a square, cf. 1 in Chapter 7 below), A4 and S4 . 1. Let α1 , . . , α4 be the roots of f in M . The cubic resolvent of f (X) is the polynomial g(Y ) = [Y − (α1 α2 + α3 α4 )] [Y − (α1 α3 + α2 α4 )] [Y − (α1 α4 + α2 α3 )].

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