- #106

martinbn

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The Galois group of a polynomial is a group of symmetries of the polynomial, which consists of all the permutations of the roots of the polynomial that can be obtained through field automorphisms.

In the case of the polynomial x^2+1, the roots of the polynomial are complex numbers i and -i, which are not real numbers. Therefore, the Galois group of the polynomial x^2+1 is a subgroup of the group of all automorphisms of the complex numbers, which is known as the Galois group of the field extension of the real numbers by the roots of the polynomial.

The Galois group of the polynomial x^2+1 is a finite group of order 2, and consists of the identity automorphism (which leaves the roots unchanged) and the automorphism which swaps the roots i and -i. This group is known as the Klein four-group, and is isomorphic to the group of symmetries of a square.

The Galois group of a polynomial is a fundamental concept in Galois theory, and is used to study the symmetries and properties of the roots of the polynomial, as well as the structure of the field extension generated by these roots.

I was reading is as it was typing it, and I thought "Damn it it is OK. I cannot complain about it." But then it wrote about Klein and the Square and I was happy again.