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#1 Help Me ! » Permuting roots... » 2009-04-03 11:47:25

ASADM
Replies: 1

Hi guys,

This question is about Galois Theory.

One of the theorems in my books states that "Any K-automorphism of the field extension L/K must keep the elements of K invariant & must permute the roots of the minimal polynomial."

I was wondering, do the K-automorphisms permutes the roots of ALL polynomials, or just the minimal polynomial?

Many thanks in advance. :-) x

#2 Help Me ! » Normal Closure.... » 2009-03-17 06:26:19

ASADM
Replies: 1

Hi guys,

I need to find the Normal Closures of:

a) A = Q(Sqrt(-5+2*Sqrt(5)))

b) B=  Q(Sqrt(1+Sqrt(5)))

both over Q.

I really don't know what it is you're supposed to do. (our lecturer is really unclear! sad )

I think it's something to doing with saying "this is a subset of that is a subset of...etc" & conjugates.

If someone could please explain I would be extremely greatful! :-)

Many thanks. x

#3 Re: Help Me ! » Finding Galois Group of Polynomials.... » 2009-03-17 06:12:46

ASADM

Don't worry - I've worked it out now! :-)

Thank you! :-D

#4 Help Me ! » Finding Galois Group of Polynomials.... » 2009-03-16 14:27:18

ASADM
Replies: 1

Hi,

I need to learn how to use resolvents, etc, to find the Galois Group of Polynomials - I'm a liiiiiiiiiiiiiiittle bit stuck at the mo! :-s (the lecturer is not being very clear at all!)

I was wondering if someone could please explain how they would apply these techniques to the following 4 equations:

1) P^4-P+1
2) P^4-3P+5
3) P^3-39P+26
4) P^4+5P^2+5

If you could also perhaps summarise the main theorems you are using/might need to use in these sorts of questions that would be fantastic! :-D But just worked solutions of the above equations would be enough. :-)

Many thanks in advance. x

#5 Re: Help Me ! » Divisibility question.... » 2009-03-16 14:14:35

ASADM

Cool - thanks mathsyperson! :-)

#6 Re: Help Me ! » Divisibility question.... » 2009-03-16 12:25:06

ASADM

.....actually, I can do the divisible by 4 bits fine, but I'm struggling with showing things are divisible by 3. :-s

Eg - how can I show that 8n^3+12n^2+6n+1 is divisible by 3? (this is x^3 for odd x. ie - x = 2n+1 )

Many thanks. x

#7 Re: Help Me ! » Divisibility question.... » 2009-03-16 11:53:39

ASADM

.....don't worry, I think I've got it now! :-) Thanks!

#8 Re: Help Me ! » Divisibility question.... » 2009-03-16 11:49:59

ASADM

Thanks! :-)

Could you please give a few more details....?

#9 Help Me ! » Divisibility question.... » 2009-03-16 10:45:03

ASADM
Replies: 8

Hi guys,

How could I go about proving that for all x>=1, x^3-x^5 is divisible by 12?

Any suggestion would be greatly appreciated! :-)

x

#10 Help Me ! » Complex Multiplication OR Toric Varieties? » 2009-03-13 01:15:23

ASADM
Replies: 1

Hi guys,

I need to pick a project for my Masters soon & I know exactly which lecturer I want to supervise me (he's AWESOME! :-D )

He's offering 2 projects that I can choose from - one of Complex Multiplication of Elliptic Curves with a little bit of Hodge Theory towards the end & one on Toric Varieties.

I've chatted with him about them, but still can't decide which to go with as he says they shall both be "very interesting" & "very fun".

Does anyone have an opinion on these 2 areas & can help me decide between them?

Any help would be most appreciated! :-)

#11 Re: Help Me ! » Galois Theory Help.... » 2009-03-06 22:50:30

ASADM

Oh yes sorry, my mistake! :-s lol! I must have missed it. Thank you very very much! :-D x

#12 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 22:19:59

ASADM

But in order to prove that JaneFairfax you would have to assume that you CAN write it in that form and then show this leads to a contradiction. (a very easy contradiction - you get that 1+Sqrt(2) = a/Sqrt(3) + [Sqrt(2)]b/Sqrt(3) for rational a &b, which is obviously impossible.)  I know it's obvious, but if yuou want to be thorough.... :-)

Could you please clarify your stuff about gcds in part (2)? :-) Thank you. x

Ricky - thanks. :-)

#13 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 13:51:21

ASADM

Well, you can't know it's irreducible unless it has no roots in the set & therefore we need to know that the obvious root ( Sqrt(6)+Sqrt(3) ) is not in the set? (unless we are just going to assume that to be the case) :-)

#14 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 13:49:16

ASADM

Thank you LadyFairfax - but I don't understand the meaning or significance of your 2nd & 3rd lines about the primitive root of unity. How do you know this gcd for k holds? And why does that lead us to conclude line 3?

Thank you very much again. :-) x

#15 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 13:38:36

ASADM

PPS - Ricky - I think I've got the answer now (7 yes?) - thankyou! :-) But do you need Gauss' Lemma? Or doesn't the irreducibility of the polynomial over Q come straight from the Eisenstein Criterion? (with p = 2)

Many thanks. :-)

#16 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 13:31:45

ASADM

PS - is there any other way to show the polynomial you arrive at in 4(a) is irreducible over Q(Sqrt(2)) other than to show that Sqrt(6)+Sqrt(3) is not an element of Q(Sqrt(2)) & hence the polynomial has no roots in Q(Sqrt(2)) ?

#17 Re: Help Me ! » Galois Theory Help.... » 2009-03-05 13:01:15

ASADM

Wow - thanks for the quick replies guys!! :-D Much appreciated!

JaneFairfax - I don't quite follow your method for 2) - Why do you look at the gcd's of n compared with 128? And why do you stop at 64? And why go up in multiples of 2?

Like I said these are LITERALLY my first worked examples, so any further explanation of the method used would be EXTREMELY appreciated! :-)

Many thanks again. x

#18 Help Me ! » Galois Theory Help.... » 2009-03-05 02:44:20

ASADM
Replies: 16

Hi guys! :-)

I'm new to the forum. Currently doing a course on Galois Theory - it's a fascinating subject but our lecturer is a bit....erm....under-par! :-s

He's only doing abstract theory & NO examples, but our exam is 100% applications! So I was wondeing if you guys could possibly help me please?

Here are a couple of questions from the last few years exam - if you have any idea how to solve ANY of them I would be EXTREMELY greatful! :-) Please could I ask you to be explicit with you methods, use of theorems, etc - these will LITERALLY be the first worked examples I've ever seen!

Thanks everyone! :-)

Questions:

1) Let the complex number A be a root of X^7+6X^2+2. Find the degree of Q(A):Q.

2) Let the complex number Zeta be a 256th root of unity. Find the degree of Q(Zeta:Q) & the minimal polynomial for Zeta over Q.

3) Let the complex number Zeta be a 12th root of unity. Find the Galois Group Gal(Q(Zeta):Q) & all subfields of Q(Zeta).

4) Find the minimal polynomial of Sqrt(6)+Sqrt(3) over (a) Q(Sqrt(2)) & (b) Q(Sqrt(3)).

Thanks so much in advance! :-) x

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