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Full-Text Articles in Physical Sciences and Mathematics
Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring
Genus Bounds For Some Dynatomic Modular Curves, Andrew W. Herring
Electronic Thesis and Dissertation Repository
We prove that for every $n \ge 10$ there are at most finitely many values $c \in \mathbb{Q} $ such that the quadratic polynomial $x^2 + c$ has a point $\alpha \in \mathbb{Q} $ of period $n$. We achieve this by proving that for these values of $n$, every $n$-th dynatomic modular curve has genus at least two.
Combinatorial Techniques In The Galois Theory Of P-Extensions, Michael Rogelstad
Combinatorial Techniques In The Galois Theory Of P-Extensions, Michael Rogelstad
Electronic Thesis and Dissertation Repository
A major open problem in current Galois theory is to characterize those profinite groups which appear as absolute Galois groups of various fields. Obtaining detailed knowledge of the structure of quotients and subgroup filtrations of Galois groups of p-extensions is an important step toward a solution. We illustrate several techniques for counting Galois p-extensions of various fields, including pythagorean fields and local fields. An expression for the number of extensions of a formally real pythagorean field having Galois group the dihedral group of order 8 is developed. We derive a formula for computing the Fp-dimension of an n-th …