What is the Galois group of a polynomial over a finite field?

Let’s learn what is the Galois group of a polynomial over a finite field. The most accurate or helpful solution is served by Mathematics.

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What is the Galois group of the splitting field of $X^8-3$ over $\mathbb{Q}$?

I've computed the splitting field of $x^8-3$ over $\mathbb{Q}$ to be $\mathbb{Q}(\sqrt[8]{3},\zeta_8)=\mathbb{Q}(\sqrt[8]{3},\sqrt{2},i)$, which is of degree 32 over $\mathbb{Q}$. The possible automorphisms are the maps fixing $\mathbb{Q}$ of form $$ \sqrt[8]{3}\mapsto \zeta_8^i\sqrt[8]{3}\quad (0\leq i\leq 7),\qquad \sqrt{2}\mapsto\pm\sqrt{2},\qquad i\mapsto\pm i. $$ There are 32 automorphisms, and thus these are all automorphisms. So I have an explicit description of the automorphisms in the...

Answer:

Your description of $G$ is perfectly fine as it is. But maybe a representation of $G$ in $GL_2(\mathbb...

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Manny Quinn at Mathematics Mark as irrelevant Undo

Other solutions

Are group theory, field theory, Galois theory or category theory useful in theoretical economics?

For example, Galois connections have made an appearance in a recent paper on implementation by Noldeke and Samuelson: Page on wiso.tu-dortmund.de.

Answer:

In the context of that paper, a "Galois connection" simply refers to an order-reversing correspondence...

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Daniel McLaury at Quora Mark as irrelevant Undo

Answer:

Klein 4-group. (This field is already Galois over the rational numbers.)

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bilolo at Yahoo! Answers Mark as irrelevant Undo

Abstract Algebra: Galois Theory construction for a polynomial HELP?

I've basically constructed the automorphisms for f(x) = (x^2 - 2)(x^2 - 3), however I need to find the Galois group. Does anyone know how to do this? I turned in my work to the professor but I wasn't sure if my solution is complete. I have the following...

Answer:

Since both Q(√2) and Q(√3) are subfields of Q(√2, √3), the automorphisms of...

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kb at Yahoo! Answers Mark as irrelevant Undo

Enter a group address in the bcc field- how can I remove only one addresss from the group addres?

I tried the answer that I was given by an other person on Yahoo. Problem is when I select my group it enters the name of the group in the BCC field and there are no individual addresses to remove. Can you give me other answers See below

Answer:

Give this yahoo tech. group a try... http://tech.groups.yahoo.com/group/Compu… Or these people...

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Lu Ann at Yahoo! Answers Mark as irrelevant Undo

Enter a group address in the bcc field- how can I remove only one address from the group addres?

For Exp. I enter a group address from my address book in the to field. Then I want to remove "Jerry" email address only because he sent me the original email. How can I remove him only and sent the email to the rest of the group

Answer:

Once you click on the group you want to blind copy, the names in the group should appear. When you see...

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Lu Ann at Yahoo! Answers Mark as irrelevant Undo

I got put in a weird awkward field trip group?

I just checked the groupings, and all of my friends literally, were put in one group and the teacher didn't include me in it..instead, he put me in a group with people i never talk to & people who freak me out a little o.0 as in, some of these people...

Answer:

Keep an open mind. Use the experience to get out of your comfort zone, you may end up making a friend...

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Krista G at Yahoo! Answers Mark as irrelevant Undo

Index of Splitting Field (galois Theory)?

Suppose that the polynomial ax^4 + bx^2 + c for some a,b,c in Q (rationals) is irreducible over Q and let K be a splitting field over Q for it. Prove that [K:Q] is either 4 or 8. I'm trying to work through the different cases of the roots of the polynomial...

Answer:

When you adjoin one root r then you get an extension of degree 4. Now either this is the splitting field...

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yummy_ea... at Yahoo! Answers Mark as irrelevant Undo

According to you, what person's/group of persons' contributions have proved so seminal in a field, whether of knowledge or art, that their work continues to define the field, even today?

Since most fields have several people whose separate contributions have shaped that field, this question is about weighing which single person out of those several had the greatest contribution according to you, and why?

Answer:

In my field, it would probably be William Kennedy Laurie Dickson, though chances are early incarnations...

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Anthony Ferreri at Quora Mark as irrelevant Undo

Is Yahoo Mail going to bring back capability to click on a group and have all names populate the "To" field?

This question is for the people who develop and release new versions of Yahoo Mail. It was so convenient in the past to be able to click on a group/category name and have everyone within that group populate the "To" field of a new email. The...

Answer:

No one directly from Yahoo ever replies- you would want to post this in Yahoo Answers Mail http://answers...

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Norman at Yahoo! Answers Mark as irrelevant Undo

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