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. / Science, Faculty of / Mathematics, Department of / Graduate
| Identifer | oai:union.ndltd.org:UBC/oai:circle.library.ubc.ca:2429/41807 |
| Date | January 1949 |
| Creators | Straight, Byron William |
| Publisher | University of British Columbia |
| Source Sets | University of British Columbia |
| Language | English |
| Detected Language | English |
| Type | Text, Thesis/Dissertation |
| Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
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