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Let F=Z_3, the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]....
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting fi...
5. Let F be a field, and let p(x) ∈ F [x] be a separable, irreducible polynomial of degree 3. Let K be the splitting field of p(x), and denote the roots of p(x) in K by α1, α2, α3. a) (10’) If char(F ) does not equal 2, 3, prove that K = F (α1 − α2).
Rings and fields- Abstract Algebra 2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of...
Rings and fields- Abstract Algebra
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q.
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
Q9 6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihed...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
Problem 1. Let G be a finite group and f : G → G a group...
Problem 1. Let G be a finite group and f : G → G a group automorphism ( isomorphism for G to G) of order 2 (i.e. f(f(x)) = x), and f has no nontrivial fixed points (i.e. f(x) = x if and only if x = 1). Prove that G is an abelian group of odd order.
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...
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group...
Let p be an odd prime. Let f(x) ∈ Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D_2p of a regular p-gon. Prove that f (x) has either all real roots or precisely one real root.
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...
Let F=Z_3 , the finite field with 3 elements. Let f(x) be an irreducible polynomial in F[x]. Let K=F[x]/(f(x)). We know that if r=[x] in K, then ris a root of f(x). Prove that f(r^3) is also a root of f(x). Which of the following are relevant ingredients for the proof? If a and b are in Z_3 then (a+b)^3=a^3+b^3 If g is an automorphism of K leaves g(r) is a root of f(x) The Remainder Theorem The Factor Theorem...
Let k be a field of positive characteristic p, and let f(x)be an irreducible polynomial. Prove that there exist an integer d and a separable irreducible polynomial fsep (2) such that f(0) = fsep (2P). The number p is called the inseparable degree of f(c). If f(1) is the minimal polynomial of an algebraic element a, the inseparable degree of a is defined to be the inseparable degree of f(1). Prove that a is inseparable if and only if its...
Rings and fields- Abstract Algebra
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g() e Fx be any polynomial. Show that every irreducible factor of f(g()) E Flx] has degree divisible by n (b) (4 points) Prove that Q(2) is not a subfield of any cyclotomic field over Q.
2. (a) (6 points) Let f (x) be an n over a field F. Let irreducible polynomial of degree g()...
Q9
6. Define Euclidean domain. 7. Let FCK be fields. Let a € K be a root of an irreducible polynomial pa) EFE. Define the near 8. Let p() be an irreducible polynomial with coefficients in the field F. Describe how to construct a field K containing a root of p(x) and what that root is. 9. State the Fundamental Theorem of Algebra. 10. Let G be a group and HCG. State what is required in order that H be...
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either all real roots precisely one real root or
2. (10) Let p be an odd prime. Let f(x) E Q(x) be an irreducible polynomial of degree p whose Galois group is the dihedral group D2p of a regular p-gon. Prove that f(x) has either...
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...
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...
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