帶參數海儂映射族群正則高度函數變異之研究

We study the iterated Galois group $G_{n}(\beta) = \gal(K_{n}(f, \beta)$ for $K_{n}(f, \beta) = K(f^{-n}(\beta))$ where $f(x)$ is the family of (finite) unicritical polynomials $f(x) = x^{d} + c \in K[x]$ with parameter $c $ ranges over a given field $K.$ Three cases are considered. Namely, (1) $K$ is a function field of finite transcendence degree over $\bar{\QQ}$ (2) $K$ is a real quadratic number field and (3) $K$ is a complete local field with a discrete valuation with residue characteristic $p > 0.$