What Does L=F(a) Indicate in Number Theory?

[Firepanda]

Wouldn't mind a hint on how to start part iii), thanks.

edit: in my notes i have for a similar question:

'L=Q(2^0.5, 3^0.5)
F=Q(6^0.5)

degree of the min polynomial = 2, because L=F(a) and [L:F]=2' (a = alpha)

Could someone clarify what L=F(a) means so I can understand the example? Thanks

 

[Firepanda]

I have found an example of how to do it here:

http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/soln6.html

Though, can someone explain to me on their example of the last line to the solution of 2.b) as to why they chose

u -i = 2^0.5

and not

u - 2^0.5 = i

or does it not matter as both give you a poly of degree 4?

 

[Petek]

The answer to the question in your second post is that you may use either expression for the reason you gave ("... both give you a poly of degree 4"). In fact, they give you the same poly. Does that clear up your original question?

Petek

 

[Firepanda]

The answer to the question in your second post is that you may use either expression for the reason you gave ("... both give you a poly of degree 4"). In fact, they give you the same poly. Does that clear up your original question?

Petek

Do they give the same poly?

If I do it that way I get

u - 2^0.5 = i

u² - 2u2^0.5 - 2 = -1

u² - 1 = 2u2^0.5

u⁴ - 2u² + 1 = 8u²

u⁴ - 10u² + 1 = 0

and their answer was

u⁴ - 2u² + 9 = 0


Hence I have two different polynomials for the question I'm trying to do, both degree 4 and monic and I'm not sure which to choose.

 

[Petek]

In your calculation, this step

u² - 2u2^0.5 - 2 = -1

is incorrect. Do you see why?

Petek