4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots. write the powers of x as x,x^2,x^3,x^4,.... e.g. x^5-2x^4-4x^3+3x^2+2x+1 5. Find the splitting field for x^2-2sqrt(2)x+3 over ℚ(sqrt(2)). Find the roots first then simplify. Note: Q(sqrt(2),u) where u is a root. Then write the simplest form of this splitting field.
Pls. answer no. 4 and 5 only.
1. Write the simplest form of the splitting field of the polynomial in no.10 using only 2 generators (take the positive conjugate). Write k,l,m,n ∈ ℚ, then write Q(k+lsqrt(u),m+nsqrt(v)) as Q(sqrt(u),sqrt(v)).
2. Find all the roots in ℂ of the polynomial x^4-4x^3+3x^2+14x+26 with one of its roots being 3+2i. List down the roots as in no.11.
[-a+sqrt(a^2-4b)]/2,[-a-sqrt(a^2-4b)]/2,[-c+sqrt(c^2-4d)]/2,[-c-sqrt(c^2-4d)]/2
So roots are 3+2i,root2,root3,root4
If the roots is not a fraction then remove the square brackets!
Simplify the roots as possible.
3. Find a monic polynomial of least degree in ℝ[x] that has 4i-1 and -3 as roots.
write the powers of x as x,x^2,x^3,x^4,.... e.g. x^5-2x^4-4x^3+3x^2+2x+1
4. Find a monic polynomial of least degree in ℝ[x] that has 1-i and 2i as roots.
write the powers of x as x,x^2,x^3,x^4,.... e.g. x^5-2x^4-4x^3+3x^2+2x+1
5. Find the splitting field for x^2-2sqrt(2)x+3 over ℚ(sqrt(2)). Find the roots first then simplify.
Note: Q(sqrt(2),u) where u is a root.
Then write the simplest form of this splitting field.
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