As briefly mentioned in the book, for polynomial equations of degree five or higher we cannot find solutions the same way that we found solutions to the equations of degree at most four.

Implicit in this statement is the notion of `solution’ that we are interested in. From the point of view of Abstract Algebra, the expressions we are looking for are what one usually calls solutions `by radicals,’ although we have not made this too precise yet. Informally, we look for a formula in which we are allowed to use:

The coefficients of the given polynomial, and

complex and real numbers,

and in which we can make use of

the elementary operations , and

radicals—i.e., we can take -th roots of any of the expressions we can obtain, for any positive integer

The amazing result that motivates our work through this course is that, indeed, no such formulas exist for polynomials of degree five or higher.

However, it is a bit misleading to read this as saying that no `formulas’ exist at all. The situation is similar to what mathematicians had to face before having the notion of complex numbers. Then, some equations could not be solved, since their solutions would involve the extraction of square roots of negative numbers. Nowadays, we understand that we can solve those equations, as long as the use of complex numbers is allowed, and we cannot otherwise.

Indeed, although no solutions by radicals are possible for the roots of quintic polynomials, if we allow a larger class of operations to be used, then solutions exist. For example, more general hypergeometric series than -th roots allow us to find the roots of the quintic. Although not nearly as popular now as they once were, hypergeometric series (a particular kind of power series) are still fairly used, for example in partition theory.

There is a well known poster from Wolfram Research explaining how Mathematica can be used to solve the quintic with the help of these functions; their webpage actually is very interesting.

Besides the usefulness of hypergeometric functions, Felix Klein found an approach using the symmetries of the Icosahedron. His book, Lectures on the Icosahedron and the Solution of Equations of the Fifth Degree, is available online for download from Cornell.

These approaches essentially propagate to higher degree equations, with some new obstacles appearing along the way, but in this sense, one can solve

any polynomial equation. The solutions are less satisfactory in that general hypergeometric functions are less well understood and much less intuitive than -th roots. For the purposes of the course, we will concentrate on solutions by radicals, since also these are the ones that lend themselves naturally to algebraic (rather than analytic) study, and the arguments work in much more generality than just over the complex numbers.

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Thursday, February 5th, 2009 at 4:13 pm and is filed under 305: Abstract Algebra I. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

I learned of this problem through Su Gao, who heard of it years ago while a post-doc at Caltech. David Gale introduced this game in the 70s, I believe. I am only aware of two references in print: Richard K. Guy. Unsolved problems in combinatorial games. In Games of No Chance, (R. J. Nowakowski ed.) MSRI Publications 29, Cambridge University Press, 1996, pp. […]

Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The problem is in the quantifiers that are implicit in the statement you are making. What you have is that for all $\epsilon>0$ and all integers $k,m$ with $k>m>0$, there is an $N$ such that if $n>N$, then $|a_n|

The relevant search term is ethnomathematics. There are several journals devoted to this topic (for instance, Revista latinoamericana de etnomatemática). Browsing them (if you have access to MathSciNet, the relevant MSC class is 01A70) and looking at their references should help you get started. Another place to look for this is in journals of history of mat […]

Some of the comments in the previous answers make a subtle mistake, and I think it may be worth clarifying some issues. I am assuming the standard sort of set theory in what follows. Cantor's diagonal theorem (mentioned in some of the answers) gives us that for any set $X$, $|X|

For $\lambda$ a scalar, let $[\lambda]$ denote the $1\times 1$ matrix whose sole entry is $\lambda$. Note that for any column vectors $a,b$, we have that $a^\top b=[a\cdot b]$ and $a[\lambda]=\lambda a$. The matrix at hand has the form $A=vw^\top$. For any $u$, we have that $$Au=(vw^\top)u=v(w^\top u)=v[w\cdot u]=(w\cdot u)v.\tag1$$ This means that there are […]

That you can list $K $ does not mean you can list its complement. Perhaps the thing to note to build your intuition is that the program is not listing the elements of $K $ in increasing order. Indeed, maybe program 20 halts on input 20 but only does it after several million steps, while program 19 doesn't halt on input 19 and program 21 halts on input 2 […]