When texts say things like this, they are just giving an informal overview of how Galois theory works, and are not being completely rigorous yet.Ĭonsider for instance the polynomial $x^2−1$. You're totally right that it doesn't follow in any obvious way. How does it follow from two examples that all possible algebraic equations involving A and B are unchanged when A and B are swapped? My problem is that they will then say 'therefore any algebraic equation with rational coefficients relating A and B is still true if A and B are swapped and the Galois group of the polynomial is a cyclic group of order 2 (since the permutations that leaves the equations unchanged are a swap and the identity)'. In either of these equations, swapping A and B gives another true equation. These roots satisfy the relations $AB=1$ and $A+B=4$. I don't understand how this generalisation is justified.įor example, consider the polynomial $x^2-4x+1=0$.
In all the texts on Galois theory I've looked at so far, when they talk about permutations of roots, they begin by giving an example of relations satisfied by the roots, show that these relations still hold after certain permutations are applied to the roots, then generalise this by saying that any relation satisfied by the roots will still hold after this permutation is applied to it.
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