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9. Let E be an extension field of a field F. (1) What does it mean...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of ...
Let E = F(a) be a (simple) extension of F. wherea E E is algebraic over F. Suppose the degree of α over F is n Then every β E E can be expressed uniquely in the form β-bo-b10 + +b-1a-1 for some bi F. (a) Show every element can be written as f (a) for some polynomial f(x) E F (b) Let m(x) be the minimal polynomial of α over F. Write m(x) r" +an-11n-1+--+ n_1α α0. Use this...
4. Let G be the Galois group of a finite field extension E of F. Let...
4. Let G be the Galois group of a finite field extension E of F. Let H and H, be subgroups of G, and let Ki and K2 be intermediate fields between F and E. For any o EG, prove that K2 = OK if and only if H2 = oHo-1,
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be se...
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable? (b) Give an example of a finite extension of fields which is inseparable. (c) Discuss the extent to which the concept of separability can be removed from a class in Galois theory.
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable?...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.
A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the gene...
A finite field is any finite extension of Fp := Z/pZ. The characteristic of a field F is the generator of the kernel of the map ι : Z → F, ι(1) = 1. (a) Prove that there exist finite fields of order pnfor any prime p. We denote such a field Fpn. (b) Prove that Fpn has characteristic p. (c) Prove that the Frobenius map φ(a) = ap is an automorphism of Fpn . (d) If f(x) ∈ Fpn...
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that...
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2 generate E is a field isomorphic to the product Gal(F/Q) x Gal(F2/Q) is X
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2...
10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). ...
Part D,E,F,G
10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Please help with the abstract algebra question detaily. Thanks. 1. Suppose r E Q. Let β cos(m). Prove that β is algebra...
Please help with the abstract algebra question detaily.
Thanks.
1. Suppose r E Q. Let β cos(m). Prove that β is algebraic over Q. Let E-Q(3). Prove that Q(3) is a normal extension of Q and that Gal(E/Q) is an abelian group.
1. Suppose r E Q. Let β cos(m). Prove that β is algebraic over Q. Let E-Q(3). Prove that Q(3) is a normal extension of Q and that Gal(E/Q) is an abelian group.
G. Shorter Questions Relating to Automorphisms and Galois Groups Let F be a field, and K a finite extension of F. Suppose a, b E K. Prove parts 1-3: 1 If an automorphism h of K fixes Fand a, then h f...
G. Shorter Questions Relating to Automorphisms and Galois Groups Let F be a field, and K a finite extension of F. Suppose a, b E K. Prove parts 1-3: 1 If an automorphism h of K fixes Fand a, then h fixes F(a). 3 Aside from the identity function, there are no a-fixing automorphisms of a(). [HINT: Note that aV2 contains only real numbers.] 4 Explain why the conclusion of part 3 does not contradict Theorem 1.
G. Shorter Questions...
2u-5 8. Let w be a root of f(x) = r +2r - 6 over the field Q. Consider z E Q(w). Find a, b, c, d e Q us + w-2 such that : a + bu + cu2 + du 9. Let E be an extension field of a field F. (1) What does it mean for an element z E E being algebraic over F? (2) What does it mean for an element z EF being transcendental...
4. Let G be the Galois group of a finite field extension E of F. Let H and H, be subgroups of G, and let Ki and K2 be intermediate fields between F and E. For any o EG, prove that K2 = OK if and only if H2 = oHo-1,
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable? (b) Give an example of a finite extension of fields which is inseparable. (c) Discuss the extent to which the concept of separability can be removed from a class in Galois theory.
Question 6 (a) What does it mean for a polynomial to be separable? What does it mean for a field extension to be separable?...
Let k C F = kla) be a simple algebraic extension. Prove that F is normal over k if and only if for every algebraic extension FC K and every o E Autk(K),(F) = F.
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2 generate E is a field isomorphic to the product Gal(F/Q) x Gal(F2/Q) is X
5. Let E be a normal extension of Q, and let Fi, F CE be two normal subex- tensionsnAssume also that Fin F2 Prove that Gal( E/O) Q and that Fi and F2...
Part D,E,F,G
10. Let p(x) +1. Let E be the splitting field for p(x) over Q. a. Find the resolvent cubic R(z). b. Prove that R(x) is irreducible over Q. c. Prove that (E:Q) 12 or 24. d. Prove: Gal(E/Q) A4 or S4 e. If p(x) (2+ az+ b)(a2 + cr + d), verify the calculations on page 100 which show that a2 is a root of the cubic polynomial r(x)3-4. 1. f. Prove: r(x) -4z 1 is irreducible in...
Please help with the abstract algebra question detaily.
Thanks.
1. Suppose r E Q. Let β cos(m). Prove that β is algebraic over Q. Let E-Q(3). Prove that Q(3) is a normal extension of Q and that Gal(E/Q) is an abelian group.
1. Suppose r E Q. Let β cos(m). Prove that β is algebraic over Q. Let E-Q(3). Prove that Q(3) is a normal extension of Q and that Gal(E/Q) is an abelian group.
G. Shorter Questions Relating to Automorphisms and Galois Groups Let F be a field, and K a finite extension of F. Suppose a, b E K. Prove parts 1-3: 1 If an automorphism h of K fixes Fand a, then h fixes F(a). 3 Aside from the identity function, there are no a-fixing automorphisms of a(). [HINT: Note that aV2 contains only real numbers.] 4 Explain why the conclusion of part 3 does not contradict Theorem 1.
G. Shorter Questions...
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