Recent content by Driessen12

you divide x^4 + x^3 + x^2 + x + 1 by x^2 because it is non zero this is valid. so we get x^2 + x + 1 + 1/x + 1/x^2. then say y = x + 1/x and plugging this back in you get y^2 + y -1 = 0 then use the quadratic formula to get (-1 + sqrt(5))/2. back to y we know that x = cos(2pi/5) + isin(2pi/5)...

touche

i also can use the fact that x^5 - 1 = 0 and i did a substitution to prove that cos(2pi/5) = (-1 + sqrt(5))/4 using the fact that x^5 - 1 + ( x - 1)(x^4 + x^3 + x^2 + x + 1) and x-1 is nonzero so the second equation on rhs has to be zero and then i used a variable substitution but i don't know...

Homework Statement Show that sin(2pi/5) = ((sqrt(10) + (2sqrt(5)))/4) Homework Equations The Attempt at a Solution i showed that cos(2pi/5) + isin(2pi/5) was a primitive 5th root of unity, and that cos(2pi/5) = (-1 + sqrt(5))/4 but i cannot figure out how to do this one any ideas?

but I think the interesting connection is that they are linearly independent and span E, making E contain u?

I think I must prove this using the idea of irreducible(minimal) polynomials, and algebraic elements

Homework Statement if F=k(u), where u is transcendental over the field k. If E is a field such that E is an extension of K and F is an extension of E, then show that u is algebraic over EHomework Equations The Attempt at a Solution i''m having trouble starting this proof, any ideas? any help...

Homework Statement show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations The Attempt at a Solution if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not...

I wasn't thinking of course, you're right. I have it all proved now

no because sin(pi) is also zero and cos(pi) is 1 giving us 1 which is exactly what I have to prove cannot happen. For example choose k to be 2 and n to be 4, then we have e^(i(pi)) which is 1. I am not sure how to prove that this cannot happen though. I need to show that it is impossible to have...

right, but to prove that z^k is not equal to zero i would have e^(i2πk/n) and this cannot equal 1 if we restrict k to be greater than or equal to 1 and less than n. So n cannot be zero and k cannot be zero, so i2(pi)k/n cannot be zero, thus z^k cannot equal 1 and is therefore primitive. and k is...

so then e^(2kpi/n) would be what i get. From there all i would need to show is that 2kpi/n cannot be zero, correct?

Homework Statement I must show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity Homework Equations a primitive root of unity is an nth root of unity that does not equal 1 when raised to the kth power for k less than n and great than or equal to 1 The Attempt at a Solution...

the field of fractions of D or field of quotients.

what would Q(D) look like?