Unformatted text preview: 150 5.25 What is the degree of the splitting fi eld of x 30 − 1 over F 5 ? Solution. Observe that x 30 − 1 = ( x 6 − 1 ) 5 in F 5 [ x ] . 5.26 Prove that if f ( x ) ∈ Q [ x ] has a rational root a , then its Galois group is the same as the Galois group of f ( x )/( x − a ) . Solution. If f ( x ) = ( x − a ) g ( x ) , then the splitting fi eld of f ( x ) over Q is the same as the splitting fi eld of g ( x ) ; hence, the Galois groups of f ( x ) and of g ( x ) are equal. 5.27 (i) Let H be a normal subgroup of a fi nite group G . If both H and G / H are solvable groups, prove that G is a solvable group. Solution. Absent. (ii) If H and K are solvable groups, prove that H × K is solvable. Solution. Absent. 5.28 We are going to improve Theorem 5.34 by eliminating the hypothesis in- volving roots of unity: if k is a fi eld and f ( x ) ∈ k [ x ] is solvable by radicals, then its Galois group Gal ( E / k ) is a solvable group....
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- Fall '11
- Group Theory, Galois theory, splitting field, Gal