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Let f(x) = F[x], let E/F be a splitting the Galois group. (i) If f(x) is irreducible, then G acts transitively on the set of all roots of f(x) (if a and Bare any two roots of f(x) in E, there exists σ € G with a (a)= B). (Hint: Lemma 50.) (ii) If f(x) has no repeated roots and G acts transitively on the roots, then f(x) is irreducible. Conclude, after comparing with Exercise 2, that irreducible polynomials are analogous to regular polygons. (Hint: If = 1; if a is a root of f(x) = g(x)h(x), then the gcd (g(x), h(x)) P(x) such that o (a) is a root of h(x), then a (a) is a common root of

Let f(x) = F[x], let E/F be a splitting the Galois group. (i) If f(x) is irreducible, then G acts transitively on the set of all roots of f(x) (if a and Bare any two roots of f(x) in E, there exists σ € G with a (a)= B). (Hint: Lemma 50.) (ii) If f(x) has no repeated roots and G acts transitively on the roots, then f(x) is irreducible. Conclude, after comparing with Exercise 2, that irreducible polynomials are analogous to regular polygons. (Hint: If = 1; if a is a root of f(x) = g(x)h(x), then the gcd (g(x), h(x)) P(x) such that o (a) is a root of h(x), then a (a) is a common root of

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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(ii) Prove = 79. Let f(x) E F[x], let E/F be a splitting field, and let G the Galois group. Gal(E/F) be (i) If f(x) is irreducible, then G acts transitively on the set of all roots of f(x) (if a and B are any two roots of f(x) in E, there exists σ € G with a (a) = 6). (Hint: Lemma 50.) (ii) If f(x) has no repeated roots and G acts transitively on the roots, then f(x) is irreducible. Conclude, after comparing with Exercise 2, that irreducible polynomials are analogous to regular polygons. (Hint: If = 1; if a is a root of f(x) = g(x)h(x), then the gcd (g(x), h(x)) g(x) such that a (a) is a root of h(x), then σ (a) is a common root of g(x) and h(x).)
Transcribed Image Text:(ii) Prove = 79. Let f(x) E F[x], let E/F be a splitting field, and let G the Galois group. Gal(E/F) be (i) If f(x) is irreducible, then G acts transitively on the set of all roots of f(x) (if a and B are any two roots of f(x) in E, there exists σ € G with a (a) = 6). (Hint: Lemma 50.) (ii) If f(x) has no repeated roots and G acts transitively on the roots, then f(x) is irreducible. Conclude, after comparing with Exercise 2, that irreducible polynomials are analogous to regular polygons. (Hint: If = 1; if a is a root of f(x) = g(x)h(x), then the gcd (g(x), h(x)) g(x) such that a (a) is a root of h(x), then σ (a) is a common root of g(x) and h(x).)
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