Visual Group Theory, Lecture 6.5: Galois group actions and normal field extensions скачать в хорошем качестве

Visual Group Theory, Lecture 6.5: Galois group actions and normal field extensions

Visual Group Theory, Lecture 6.5: Galois group actions and normal field extensions If f(x) has a root in an extension field F of Q, then any automorphism of F permutes the roots of f(x). This means that there is a group action of Gal(f(x)) on the roots of f(x), and this action has only one orbit iff f(x) is irreducible. An extension of Q is said to be "normal" if it is the splitting field of some polynomial, and the degree of a normal extension of the order of its Galois group. We ilustrate these concept with several examples: the reducible polynomial x^4-5x^2+6, and the irreducible polynomial x^3-2. Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...