Can an irrational number be constructed which is a) not any known transcendental number b) not a surd? If yes, then how can I construct one? A detailed answer regarding the theory behind this and some references will be appreciated. I modified the a) part to what it is now, because I am guessing numbers are either algebraic or not algebraic(I.e. transcendental). Is this correct? I am thinking along the lines of constructing a sequence which converges to the desired number, but then how to construct a sequence to a desired limit?
2026-04-03 16:24:26.1775233466
Construction of irrational numbers
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The answer is yes. If you know about Galois theory, you need an extension of $\mathbb{Q}$ that has a non-solvable Galois group (like $S_5$). If you don't know about Galois theory, then the roots of the polynomial $x^5-80x+5$ are irrational numbers but they are not surds and not transcendental.