Titlebook: Excursions in Number Theory, Algebra, and Analysis; Kenneth Ireland,Al Cuoco Textbook 2023 The Editor(s) (if applicable) and The Author(s)

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Introduction Foundations Research,r, there are many numbers that are not algebraic over .. In fact, the algebraically closed field of all algebraic numbers in . is a countable set, for you can check that the algebraic numbers over . that have a minimal polynomial of degree . are countable. Letting . vary gives a countable collection of countable sets, which is therefore countable.

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Dialing In Problems,ndem with) your formal instruction or reading. They cover a wide range of topics. Some of them will not be familiar to you. But try them now, look things up (in this book, for example), and come back to them as you proceed through the text.

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Irrational, Algebraic, and Transcendental Numbers,r, there are many numbers that are not algebraic over .. In fact, the algebraically closed field of all algebraic numbers in . is a countable set, for you can check that the algebraic numbers over . that have a minimal polynomial of degree . are countable. Letting . vary gives a countable collection