MATH 422 Midterm: MATH 422+501 2006 Winter Test 1
Document Summary
De ne the following terms: (a) centre of a group (b) normal subgroup (c) solvable group (d) sylow subgroup (e) minimal polynomial (f) galois extension (g) separable polynomial. Carefully state: (a) the orbit equation, (b) a result describing the orbits of an action of the galois group of the splitting. Eld of a polynomial, (c) a criterion by which to recognize galois extensions. Let (z/nz) be the multiplicative group of the ring z/nz. In other words, (z/nz) is the group of residue classes a + nz modulo n, such that a is coprime to n, with multiplication as group operation. Let g be a cyclic group of order n. prove that. Prove that the elds q( 2) r and q( 3) r are not isomorphic. Let k be a eld and assume that the characteristic of k does not divide the integer n > 0. Let l/k be a splitting eld of the polynomial xn 1 k[x].
