24.To which permutation in S_4 does the automorphism (sigma tau) στ correspond to based on its action on the roots of f above? Write as a product of transpositions. A transposition that switches two elements x,y of {1,2,3,4} should be typed as (xy) without spaces in between!
Pls. answer no. 24 and 25 only.
21.Let K be the splitting field of x^23 - 1 over ℚ and zeta ζ be a 23rd primitive root of unity. Then what is order of the Galois group of K over ℚ?
22.How many irreducible factors over over ℚ does x^48 - 1 have?
23.The splitting field of f=(x^2-2)(x^2+1) over ℚ is ℚ(sqrt(2),i) because its roots are sqrt(2),-sqrt(2),i,-i. The Galois group of f contains the following automorphisms: sigma σ: sqrt(2) ↦ -sqrt(2) and tau τ: i ↦ -i. Which subfield of the splitting field is fixed by sigma?
24.To which permutation in S_4 does the automorphism (sigma tau) στ correspond to based on its action on the roots of f above? Write as a product of transpositions. A transposition that switches two elements x,y of {1,2,3,4} should be typed as (xy) without spaces in between!
25.The field Z_2(u) is a splitting field for f(x)=x^3+x+1 over Z_2. This polynomial is separable. If u is a root, then find the two other roots of f(x) as powers of u.
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