The Embedding Problem in Galois Theory by B. B. Lure, and D. K. Faddeev V. V. Ishkhanov

By B. B. Lure, and D. K. Faddeev V. V. Ishkhanov

The relevant challenge of recent Galois concept comprises the inverse challenge: given a box $k$ and a gaggle $G$, build an extension $L/k$ with Galois staff $G$. The embedding challenge for fields generalizes the inverse challenge and is composed to find the stipulations below which you can actually build a box $L$ basic over $k$, with staff $G$, such that $L$ extends a given basic extension $K/k$ with Galois workforce $G/A$. furthermore, the necessities utilized to the item $L$ to be stumbled on are typically weakened: it isn't useful for $L$ to be a box, yet $L$ needs to be a Galois algebra over the sphere $k$, with crew $G$. during this atmosphere the embedding challenge is wealthy in content material. however the inverse challenge by way of Galois algebras is negative in content material simply because a Galois algebra offering an answer of the inverse challenge continuously exists and should be simply built. The embedding challenge is a fruitful method of the answer of the inverse challenge in Galois concept. This booklet relies on D. okay. Faddeev’s lectures on embedding conception at St. Petersburg collage and comprises the most effects at the embedding challenge. All levels of improvement are provided in a methodical and unified demeanour.

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The matrices T9 constructed in the preceding section have the properties: T 9 t = T 9µ(t) and T9192 = Tf(T92 c91 ,92 , where c91 ,92 is obtained by lifting of a cocycle of Z 1(Fe,K*) to the group Ge. Let 89 = detT9 EK*, then 89192 = oz;o92 c~ 1 , 92 and 09t = detT9t = det(T9µ(t)) ~ 89µo(t). Let gl = 71 ti, g2 = 72t2, where Ji, h E Fe, 71 , 72 are the preimages of them in Ge, ti E N. Then 89192 = 87 1 t 1 72 t 2 = 7 = , where his the cocycle of the extension Ge fihh(f1,h)t 12 t2 fihµoh(f1,f2)µo(t 72 1 t2) of the group N by Fe.

Hence Cg 1t 1 ,g 2h = Cg1,u2· Now let q9 = µ- 1(T9) E K[N]e and w9 = u 9q; 1. Then Wgt = Ugtq;/ = UgUt(µ- 1(T9µ(t)))-l = UgUtUt 1q; 1 = Wg, so that the w9 depend only on cp(g). / belongs to K[N]e and its image under the map µonto Kd is equal to T 9192 T 9-; 1(T9-;: 1)92 T 92 T 9-; 1 = T 9192 T 9-; 1T 9-;: 92 = c91 ,92 . Hence, w91 w92 = w9192 c9192 . Therefore, c9192 is a two-dimensional cocycle of the group Fe with values in K*. The elements w 9 belong to the algebra eAe because g E Ge· Let us establish the commutation relations of these elements with the elements of the algebra K[N]e.

The exactness of the upper row and the commutativity of the diagram are obvious. Now we prove uniqueness. Let a commutative diagram (*) with exact rows be given. We map the elements 9 E G to the pairs A(9) = ('1/J(9), cp(9)) E G 1 x F. By the commutativity of the diagram, we have cp 11/J(9) = Ocp(9), so that A(9) is contained in the direct product of G 1 and F with amalgamated quotient group F 1 . The fact that the mapping A is a homomorphism is obvious. The kernel of A is the set of 9 E G such that '1/J(9) = 1 and cp(9) = 1.

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