11.Find a splitting field of x^4-4x^2-5 over ℚ. Write the splitting field as a finitely generated extension field generated by some algebraic elements over ℚ 12.What is the order of a nonidentity automorphism in the Galois group of the polynomial x^4-4x^2-5? 13.How many subfields of degree 2 over ℚ does this splitting field in no.7 have?
Pls. answer no. 11,12 and 13 only thank you.
11.Find a splitting field of x^4-4x^2-5 over ℚ. Write the splitting field as a finitely generated extension field generated by some algebraic elements over ℚ
12.What is the order of a nonidentity automorphism in the Galois group of the polynomial x^4-4x^2-5?
13.How many subfields of degree 2 over ℚ does this splitting field in no.7 have?
14.Factor x^4+2x^3-8x^2-6x-1 into two irreducible quadratic polynomials over ℚ.
Hint: Attempt to factor in the form (x^2+ax+b)(x^2+cx+d) where a < c.
15.Find the roots of the quadratic factors above and write the roots of x^2+ax+b first. Since the roots are conjugates, write the roots as follows:
[-a+sqrt(a^2-4b)]/2,[-a-sqrt(a^2-4b)]/2,[-c+sqrt(c^2-4d)]/2,[-c-sqrt(c^2-4d)]/2
If the roots is not a fraction then remove the square brackets!
Simplify the roots as possible.
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