# Graduate Thesis Or Dissertation

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• If P is an integer polynomial denote the degree of P by ∂(P) and let H(P) be the maximum of the absolute value of the coefficients of P. Define Λ(P)=2[superscript ∂(P)]H(P) and for a fixed prime p let C[subscript p] denote the completion of the algebraic closure of the p-adic numbers. We generalize the order function of Mahler to the p-adic numbers by associating each θ∈C[subscript p] to the function O(u|θ) = max log (1/|P(θ)|), where |*| denotes the p-adic absolute value and the maximum is taken over all integer polynomials P(x) ∈ Z[x] satisfying Λ(P) ≤ u and P(θ) ≠ 0. Similarly, define O*(u|θ) = max log (1/|θ-α|) where the maximum is now taken over the set of all algebraic numbers α ≠ θ with minimal polynomial m satisfying Λ(m) ≤ u. Placing a partial order >> and equivalence relation ≍ on both O and O* induces two corresponding partial orders and equivalence relations on Cp. In this thesis we demonstrate several results concerning O and O*. In particular, it is proved that if θ is algebraic over Qp then θ must satisfy O*(u|θ) >> log u and if θ,η ∈ Cp that are algebraically dependent over the Q then O(u|θ) ≍ O(u|η). Also, under certain conditions, given a function g we construct θ ∈ Cp such that g(c'(log log u)[superscript ½) << O*(u|θ) and O*(u|θ) << g(c log log u), where c and c' are positive constants. The transcendence type considers the limiting behavior of O and O* and will be used to prove that under certain conditions O and O* behave similarly. Finally, given τ ≥ (3+√5)/2 we demonstrate that it is possible to construct elements of Cp with transcendence type equal to τ.
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• description.provenance : Approved for entry into archive by Julie Kurtz(julie.kurtz@oregonstate.edu) on 2009-06-16T17:31:34Z (GMT) No. of bitstreams: 1 Dietelthesis09.pdf: 446804 bytes, checksum: b52937c22bb54e0563de51dac7c60250 (MD5)
• description.provenance : Approved for entry into archive by Laura Wilson(laura.wilson@oregonstate.edu) on 2009-06-18T21:46:26Z (GMT) No. of bitstreams: 1 Dietelthesis09.pdf: 446804 bytes, checksum: b52937c22bb54e0563de51dac7c60250 (MD5)
• description.provenance : Submitted by Brian Dietel (dietelb@onid.orst.edu) on 2009-06-09T16:43:27Z No. of bitstreams: 1 Dietelthesis.pdf: 446806 bytes, checksum: c7248ab79afc01766221646b961d87d9 (MD5)
• description.provenance : Rejected by Julie Kurtz(julie.kurtz@oregonstate.edu), reason: Rejecting so you can change the commencement date on the title page to - June 2009. Once revised, open the item that was rejected, replace the attached file with the revised file and resubmit. Thanks, Julie on 2009-06-11T17:24:17Z (GMT)
• description.provenance : Submitted by Brian Dietel (dietelb@onid.orst.edu) on 2009-06-11T20:03:55Z No. of bitstreams: 1 Dietelthesis09.pdf: 446804 bytes, checksum: b52937c22bb54e0563de51dac7c60250 (MD5)
• description.provenance : Made available in DSpace on 2009-06-18T21:46:26Z (GMT). No. of bitstreams: 1 Dietelthesis09.pdf: 446804 bytes, checksum: b52937c22bb54e0563de51dac7c60250 (MD5)

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