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The book Calcul mathématique avec Sage was translated into German as Rechnen mit Sage and into English as Computational mathematics with SageMath.

Section 5.3 of the English translation ends like this:

Algebraic Numbers. An algebraic number is defined as root of a polynomial. When the polynomial degree is 5 or more, in general it is not possible to explicitly write its roots in terms of the operations $+$, $−$, $\times$, $/$, $\sqrt{}$. However, many computations involving the roots can be performed successfully without any other information than the polynomial itself.

sage: k.<a> = NumberField(x^3 + x + 1); a^3; a^4+3*a
-a - 1
-a^2 + 2*a

This book does not describe in detail how to play with algebraic numbers in Sage, however several examples can be found in Chapters 7 and 9.

In general I would recommend this book as a very good introduction to computer algebra and numerical computation as well as to SageMath, with a lot of examples.

(Disclaimer: I know some of the authors.)