UBC Theses and Dissertations
On the Galois groups of certain algebraic number fields Straight, Byron William
This thesis is concerned with the Galois groups of the root fields of the equations x[superscript]P - a = 0, (x[superscript]p - a)•(x[superscript]q - b) = 0 and (x[superscript]q - b) [superscript]p - a = 0, where p and q are distinct primes, and a and b are rationals. The correspondence of subflelds and subgroups is studied for each of the three cases. The field [formula omitted] formed by adjoining to the rational field F the elements [formula omitted and ⍺, a primitive pth root of unity, is shown to be the root field of x[superscript]p - a = 0, normal over F of degree p(p-l). The Galois group of [formula omitted] over F Is found to be the metacyclic group constructed from generators s and t subject to relations s[superscript]p = 1, t[superscript]p ⁻¹ = 1 and st = ts[superscript]r, where r is a primitive root modulo p, and where s is the automorphism which maps [formula omitted] onto [formula omitted] while t is the automorphism which maps ⍺. onto ⍺ [superscript]r. Various subgroups and corresponding subflelds are studied and nine theorems proven on their correspondences, illustrated with a partial lattice diagram. The field [formula omitted]where β is a primitive qth root of unity is shown to be the root field of (x[superscript]p - a)-(x[superscript]q - b) = 0 and the Galois group is proven to be the direct product of two of the type for the field [formula omitted]. The field [formula omitted] for i = 1, 2, 3 ... p, which is the root field of the equation (x[superscript]q - b) [superscript]p - a = 0 is studied and shown to have degree pq[superscript]p-(p-1)•(q-1). The Galois group is found to be generated by four independent generators: s, t, u, v subject to eleven defining relations. Here the elements s, t, u, v are the automorphisms which respectively map [formula omitted] onto [formular omitted], ⍺ onto ⍺[superscript r, [formula omitted] onto [formula omitted] β onto β [superscript]where w is a primitive root modulo q. A partial lattice diagram illustrates the correspondence of subgroups and subflelds. The thesis was carried out under the supervision of Dr. D. C. Murdoch.
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