Problem 5. Give an example of an irreducible polynomial f ∈ K[x] which has roots a,...
Problem 5. Give an example of an irreducible polynomial f ∈ K[x] which has roots a, b, c in its splitting field such that the fields K(a, b) and K(a, c) are not isomorphic over K.
Homework Answers
Add Answer to:
Problem 5. Give an example of an irreducible polynomial f ∈ K[x]
which has roots a,...
Show that the irreducible polynomial x4 - 2 over Q, has roots a, b, c in its splitting field such...
Show that the irreducible polynomial x4 - 2 over Q, has roots a, b, c in its splitting field such that the fields Q(a, b) and Q(a, c) are not isomorphic over Q (Hint: The roots are (4√2, -4√2, 4√2i, -4√2i), and the splitting field is Q(4√2, i,).)
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).
The polynomial 23 - 2 + 1 has no roots in Zg, so it is irreducible...
The polynomial 23 - 2 + 1 has no roots in Zg, so it is irreducible in Zg[] (you don't have to show this). Suppose a is a root of 23 - 2 + 1 in an extension of Zz 1. Show that a +1 and a + 2 are also roots of 23 - 2+1 Conclude that Zz(a) is the splitting field of 23 - 2+1, and thus a Galois extension of Zz. (Hint: Theorem 3 from Chapter 20...
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition se...
just part c and d
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition series. Explain why your example works. (b) Find a refinement of the series you gave above which is a composition series. That is, show how to extend your series into a composition series. (c) Find all other composition series and briefly explain how you know you have them all. (d) If p(x) is a polynomial whose splitting...
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()...
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...
Polynomial over the Fields: a) If p(x) an element of F[x] is not irreducible, then there are at least two polynomials g(x) and h(x), neither which is a constant, such that p(x)=g(x)h(x). Explain b) U...
Polynomial over the Fields: a) If p(x) an element of F[x] is not irreducible, then there are at least two polynomials g(x) and h(x), neither which is a constant, such that p(x)=g(x)h(x). Explain b) Use problem a to prove: If p(x) is not irreducible, then p(x)=j(x)k(x), where both j(x) and k(x) are polynomials of lower degree than p(x).
Theorem 14.7. If f(x) € R[x] is an irreducible polynomial, then deg(f(x)) is either 1 or...
Theorem 14.7. If f(x) € R[x] is an irreducible polynomial, then deg(f(x)) is either 1 or 2. We can determine which quadratic polynomials in R[x] are irreducible by using the quadratic formula and checking for real roots. Activity 14.8. Factor f(x) = 2 – 4.x in R[2] into a product of irreducible polynomials in R[2].
Preview Activity 14.1. In previous investigations, we defined irreducible polynomials and showed that irreducible polynomials in...
Preview Activity 14.1. In previous investigations, we defined irreducible polynomials and showed that irreducible polynomials in polynomial rings over fields play the same role as primes play in Z. In this investigation we will explore some methods to determine when a polynomial is irreducible, with a special emphasis on polynomials with coefficients in C, R, and Q. To begin, we will review the definition and a simple case. Let F be a field. (a) Give a formal definition of what...
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 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 (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
The polynomial 23 - 2 + 1 has no roots in Zg, so it is irreducible in Zg[] (you don't have to show this). Suppose a is a root of 23 - 2 + 1 in an extension of Zz 1. Show that a +1 and a + 2 are also roots of 23 - 2+1 Conclude that Zz(a) is the splitting field of 23 - 2+1, and thus a Galois extension of Zz. (Hint: Theorem 3 from Chapter 20...
just part c and d
3. Consider the group Z45 (a) Give an example of a subnormal series which is not a composition series. Explain why your example works. (b) Find a refinement of the series you gave above which is a composition series. That is, show how to extend your series into a composition series. (c) Find all other composition series and briefly explain how you know you have them all. (d) If p(x) is a polynomial whose splitting...
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()...
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...
Theorem 14.7. If f(x) € R[x] is an irreducible polynomial, then deg(f(x)) is either 1 or 2. We can determine which quadratic polynomials in R[x] are irreducible by using the quadratic formula and checking for real roots. Activity 14.8. Factor f(x) = 2 – 4.x in R[2] into a product of irreducible polynomials in R[2].
Preview Activity 14.1. In previous investigations, we defined irreducible polynomials and showed that irreducible polynomials in polynomial rings over fields play the same role as primes play in Z. In this investigation we will explore some methods to determine when a polynomial is irreducible, with a special emphasis on polynomials with coefficients in C, R, and Q. To begin, we will review the definition and a simple case. Let F be a field. (a) Give a formal definition of what...
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 (ab)^3=(a^3)(b^3) The Remainder Theorem If a and b are in Z_3 then (a+b)^3=2^3+b^3 For all a in Z_3, a^3=a The first isomorphism...
Most questions answered within 3 hours.
-
Work of 1950 J is done by stirring a perfectly insulated beaker
containing 75 g of...
asked 19 minutes ago -
The neighborhood kids set up an outdoor lemonade stand in
Maryland in June. They find that...
asked 20 minutes ago -
9. A company has a beginning inventory of 4,000 units. The
company estimates it will sell...
asked 34 minutes ago -
A patient goes to the doctor's office with symptoms of a urinary
tract infection and provides...
asked 36 minutes ago -
When responding to the essay questions, be sure to cite any
material you obtained from a...
asked 36 minutes ago -
The energy of an electron in a 2.25-eV-deep potential well is
1.50 eV.At what distance into...
asked 38 minutes ago -
Q1:Which three evolutionary innovations are present in land
plants (but not all land plants) that allowed...
asked 41 minutes ago -
Lymphosarcoma is
extremely rare. Risk factors for the disease are largely unknown.
What kind of study...
asked 43 minutes ago -
The solubility of benzoic acid in water is:
0.29g/100mL at 20°C
6.8g/100mL at 100°C
a) What...
asked 44 minutes ago -
Which food law was passed in 1996 and changed how pesticide
residues on food were regulated...
asked 1 hour ago -
companies either hire outside programmers to
write_____ software or use their own internal developers.
asked 1 hour ago -
A magnetic dipole m(t) = m_0*cos(ωt) can be
described as current density j(r,t) = −cm(t) ×...
asked 1 hour ago