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Fields 12. Show that the points of intersection of a circle in the plane of a field F and a line in the plane of F are points in the plane of F or in the plane of F(Va), where a E F and a is positive. Give an example of a circle and a line in the plane of Q whose points of intersection are not in the plane of Q. 13. Prove that 8x3 - 6x - 1 is irreducible over Q. 14. Use the fact that 8 cos (217/7) + 4 cos2(217/7) - 4 cos(271/7) - 1 = 0 to prove that a regular seven-sided polygon is not constructible with an unmarked straightedge and a compass. 15. Show that a regular 9-gon cannot be constructed with an unmarked straightedge and a compass. 16. Show that if a regular n-gon is constructible, then so is a regular 2n-gon. 17. (Squaring the Circle) Show that it is impossible to construct, with an unmarked straightedge and a compass, a square whose area equals that of a circle of radius 1. You may use the fact that 7 is transcendental over Q. 18. Use the fact that 4 cos2(217/5) + 2 cos(271/5) - 1 = 0 to prove that a regular pentagon is constructible. 19. Can the cube be "tripled"? 20. Can the cube be "quadrupled"? 21. Can the circle be "cubed"? 22. If a, b, and c are constructible, show that the real roots of ax2 + bx + c are constructible