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# algebraic conjugates

Let $L$ be an algebraic extension of a field $K$, and let $\alpha_{1}\in L$ be algebraic over $K$. Then $\alpha_{1}$ is the root of a minimal polynomial $f(x)\in K[x]$. Denote the other roots of $f(x)$ in $L$ by $\alpha_{2}$, $\alpha_{3},\ldots,\alpha_{n}$. These (along with $\alpha_{1}$ itself) are the *algebraic conjugates* of $\alpha_{1}$ and any two are said to be *algebraically conjugate*.

The notion of algebraic conjugacy is a special case of group conjugacy in the case where the group in question is the Galois group of the above minimal polynomial, viewed as acting on the roots of said polynomial.

Related:

ComplexConjugate, ConjugationMnemonic

Synonym:

algebraically conjugate, conjugate

Type of Math Object:

Definition

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

11R04*no label found*

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## Attached Articles

## Corrections

Related, generality, names by bbukh ✓

broken by yark ✓

other synonyms by Wkbj79 ✓

alpha1 is conjugate of itself by pahio ✓

broken by yark ✓

other synonyms by Wkbj79 ✓

alpha1 is conjugate of itself by pahio ✓

## Comments

## Obtuse Definition

Sorry, but the description here provides no useful info. The expected users of this definition would be high school students. I have an advanced degree and it makes no sense to me so it is useless to other possible users.

If someone truly believes that this extreme abstraction is useful, then put it under a subheading labeled advanced definition.

A clear definition with examples is available at http://www.math.utah.edu/online/1010/complex/ under division.

## Re: Obtuse Definition

I've never known a high-school student that I would expect to understand this definition because it's not intended for high-school students. You are confusing the notion of the conjugate of a complex number with conjugates in the sense of Galois Theory. These are definitely not the same thing. Moreover, this is not a case of "extreme abstraction," but a standard definition in advanced undergraduate\beginning graduate math curriculum.

## Re: Obtuse Definition

the definition seems very concise and clear to me. I think you're confusing algebraic conjugates with complex conjugates, which are related, but much simpler and more restrictive.

## Re: Obtuse Definition

How about including some of azdbacks4234's comments as a header to indicate in a general fashion what the entry is about? His comments need a few more words on the matter to be complete. A cross reference to a more general entry would also be an improvement.

From: the first paragraph of the main entry of the website:

"PlanetMath is a virtual community which aims to help make mathematical knowledge more accessible."

## Re: Obtuse Definition

See the entry:

http://planetmath.org/encyclopedia/ComplexConjugate.html

The entry under discussion is not about complex conjugation...that's why the term doesn't appear in the title. If you aren't trained in mathematics you can't expect to understand everything on this site. Having an "advanced degree" in one field doesn't mean you're naturally going to understand everything in some other field...that's why there are Ph.D's in math, and there are Ph.D's in physics...they are completely different subjects. A certain amount of background is necessary just to understand some definitions, and it would be severely inefficient to attempt to provide all this background in an entry which is meant just as a definition. That's why we have the linking system. Moreover, this IS the general entry. The notion of a complex conjugate is subsumed by that of an algebraic conjugate, as gel alluded to. Making mathematics more accessible doesn't mean making advanced mathematics comprehensible to everyone; if this were possible, we'd have no need for graduate school because everybody could be a mathematician. You can't explain modern algebraic geometry to a high-school student in a useful way unless that high-school student is extraordinarily gifted in mathematics. The site makes information about all facets of mathematics available to everyone on the internet...the necessary background for understanding a given entry is implicit in the subject matter of the entry, and if that necessary background is unclear to you, it's probably likely you don't have it.

## Re: Obtuse Definition

If it is a general entry, then it certainly needs a few intro words to provide a general description. A couple of sentences along the lines of azdbacks4234's prior post with gel's post would provide an excellant header. Noting that this entry is a generalization of complex conjugate would be especially informative.

## Re: Obtuse Definition

Alright, just one more thing. If you feel like an entry needs something, file a correction to it. This way your concerns are brought to the attention of the entry's owner. The link to do this is at the bottom of the entry page. Note that I am neither endorsing nor dismissing your assertion that anything needs to be changed with this entry, I'm just trying to point you in the right direction as a new user of the site. I would caution you, however, to take a look at some other entries on the site, at various levels of mathematical sophistication, to get an idea about how mathematical writing looks. I reiterate that the definition in this entry is basic for senior undergraduate and beginning graduate math students. To make it fully comprehensible for a reader with no mathematical background would be absurd, requiring at least explanations of all of the following terms:

group

Galois group

field

field extension

algebraic element

algebraic field extension

minimal polynomial

group conjugacy

Trust me when I tell you these terms cover a wide range of mathematical material, far more than could be fit reasonably into a single entry. Math is very much cumulative.

## Re: Obtuse Definition

Good day.

> the description here provides no useful info.

It does.

It just does not give information on the subject you expect.

> The expected users of this definition would be high school

> students.

No. The replies of other users should have clarified this.

> I have an advanced degree

In what?

> and it makes no sense

> to me so it is useless to other possible users.

True but not so relevant to justify editing the entry.

> If someone truly believes that this extreme abstraction is

> useful, then put it under a subheading labeled advanced

> definition.

There is no need to.

Anyone might search "conjugate" with PM search engine, and get

all the possible entry. At this point, however, it is his/her

duty to understand which suits his/her needs.

> A clear definition with examples is available at

> http://www.math.utah.edu/online/1010/complex/ under

> division.

Thank you very much for signaling.

Best regards,

Silvio