By M. M. Postnikov

The 1st half explores Galois thought, concentrating on comparable ideas from box concept. the second one half discusses the answer of equations through radicals, returning to the overall conception of teams for appropriate proof, studying equations solvable through radicals and their building, and concludes with the unsolvability by way of radicals of the final equation of measure n is larger than 5. 1962 variation.

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Thus the order of the subgroup G' is equal to the order of the whole group G(L, P). Thus the order of the subgroup G' is equal to the order of the whole group G(L, P), whence it follows that G' = G(L, P). Thus it has been proved that the mapping (1) is epimorphic. The mapping induced by the homomorphism is, hence, an isomorphism of the factor group G(K, P)/G(K, L) onto the group G(L, P). Thus : GALOIS THEORY 41 the Galois group of a normal intermediate field L over the field P is isomorphic to the factor group of the Galois group of the field K over the field P by the Galois group of the field K over the field L.

The Galois group of the field K over the field L is a subgroup of the Galois group of the field K over the field P. Its order is equal to the degree [K: L] of the field K over the field L. Now let H be an arbitrary subgroup of the Galois group G(K, P). It is obvious that the set of all elements of the field K, left invariant by any automorphism from the subgroup H, is a subfield of the field K. e. is an intermediate field. We will denote it by K(G, H). ,T. be all the elements of the subgroup H (thus m is the order of the subgroup H).

Its order is equal to the degree [K: L] of the field K over the field L. Now let H be an arbitrary subgroup of the Galois group G(K, P). It is obvious that the set of all elements of the field K, left invariant by any automorphism from the subgroup H, is a subfield of the field K. e. is an intermediate field. We will denote it by K(G, H). ,T. be all the elements of the subgroup H (thus m is the order of the subgroup H). We consider the polynomial h(x) = fl (x - 0T'). i=1 36 FOUNDATIONS OF GALOIS THEORY Its roots are the numbers OT' = 0, 0T2, ...