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Polynomial and exponential equations modulo primes

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Title: Polynomial and exponential equations modulo primes
Author(s): Järviniemi, Olli
Contributor: University of Helsinki, Faculty of Science
Degree program: Master's Programme in Mathematics and Statistics
Specialisation: Mathematics
Language: English
Acceptance year: 2021
Abstract:
This thesis is motivated by the following questions: What can we say about the set of primes p for which the equation f(x) = 0 (mod p) is solvable when f is (i) a polynomial or (ii) of the form a^x - b? Part I focuses on polynomial equations modulo primes. Chapter 2 focuses on the simultaneous solvability of such equations. Chapter 3 discusses classical topics in algebraic number theory, including Galois groups, finite fields and the Artin symbol, from this point of view. Part II focuses on exponential equations modulo primes. Artin's famous primitive root conjecture and Hooley's conditional solution is discussed in Chapter 4. Tools on Kummer-type extensions are given in Chapter 5 and a multivariable generalization of a method of Lenstra is presented in Chapter 6. These are put to use in Chapter 7, where solutions to several applications, including the Schinzel-Wójcik problem on the equality of orders of integers modulo primes, are given.
Keyword(s): Prime divisors of polynomials Chebotarev's density theorem Artin's primitive root conjecture


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