Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 3431. Show that each of the following numbers is algebraic over Q by finding theminimal polynomial of the number over Q.√(a) 1/3 + √ 7(b) √ 3 + 3√ 5(c) √ 3 + √ 2 i(d) cos θ + i sin θ for θ = 2π/n with n ∈ N(e) √ 3 √ 2 − i2. Find a basis for each of the following field extensions. What is the degree ofeach extension?(a) Q( √ 3, √ 6 ) over Q(b) Q( 3√ 2, 3√ 3 ) over Q(c) Q( √ 2, i) over Q(d) Q( √ 3, √ 5, √ 7 ) over Q(e) Q( √ 2, 3√ 2 ) over Q(f) Q( √ 8 ) over Q( √ 2 )(g) Q(i, √ 2 + i, √ 3 + i) over Q(h) Q( √ 2 + √ 5 ) over Q( √ 5 )(i) Q( √ 2, √ 6 + √ 10 ) over Q( √ 3 + √ 5 )3. Find the splitting field for each of the following polynomials.(a) x 4 − 10x 2 + 21 over Q(c) x 3 + 2x + 2 over Z 3(b) x 4 + 1 over Q(d) x 3 − 3 over Q4. Determine all of the subfields of Q( 4√ 3, i).5. Show that Z 2 [x]/〈x 3 + x + 1〉 is a field with eight elements. Construct amultiplication table for the multiplicative group of the field.6. Show that the regular 9-gon is not constructible with a straightedge andcompass, but that the regular 20-gon is constructible.7. Prove that the cosine of one degree (cos 1 ◦ ) is algebraic over Q but not constructible.8. Can a cube be constructed with three times the volume of a given cube?9. Prove that Q( √ 3, 4√ 3, 8√ 3, . . .) is an algebraic extension of Q but not a finiteextension.10. Prove or disprove: π is algebraic over Q(π 3 ).