# Can polynomials have solutions other than numbers?

We have learned in our previously classes that whenever we have a polynomial function then it has values of $x$, which satisfy . Are there other structures like Matrix, Vectors or something else which satisfy a polynomial? If so then kindly tell.

• If $f(x) = a_m x^m + \cdots + a_1 x + a_0$ is a polynomial with coefficients in a field $F$, you can plug a square matrix $A \in F^{n \times n}$ into this polynomial to obtain the matrix $f(A) = a_m A^m + \cdots + a_1 A + a_0 I$. – littleO Dec 21 '17 at 12:27

A very famous theorem is the Cayley-Hamilton theorem about the characteristic polynomial. It states that a square matrix $A$ does fulfil its own characteristic polynomial for the eigenvalues.

• Thank you so much ... – abstract Dec 21 '17 at 12:30
• Yes this is really good example. It shows that the field of matrices over any field always have larger set of solutions than their scalar field. – mathreadler Dec 21 '17 at 12:32
• And , let me ask something more , are there other than matrices as solutions if so please make me know those too – abstract Dec 21 '17 at 12:33
• It is possible to show that any algebraic group can be written as matrices of elements from less complicated fields (if matrix multiplication is to "do the same thing" to the group element as the group operation does). So in some sense you can build arbitrarily complicated "numbers" with matrices. – mathreadler Dec 21 '17 at 12:39

Sure there is. Whenever you have a structure with a multiplication operation and an addition operation that interacts relatively nicely ("ring" is the archetypal definition of "nicely", but you can go somewhat further with less strict requirements if you want), you can make polynomials.

For instance, it is a known result that each matrix is a root of its own characteristic polynomial.

An important part of modern number theory concerns solutions to polynomial equations in the integers modulo prime numbers. For instance, the modularity theorem, famous for being the last building block in the final proof of Fermat's last theorem, was first conceived because it was noticed that the number of solutions of certain types of polynomial equations modulo any prime $p$ was tightly related to the degree $p$ coefficient in the expansion of certain (previously unrelated) functions.

• Wow !! I have been unaware of these things thanx , I would like to learn more from you if possible . – abstract Dec 21 '17 at 12:35

In my opinion polynomials are defined on symbols and we can relate values to these symbols. But they are still symbols nonetheless. Look at this for an example where the point is not to solve the polynomial for a ceertain number rather just to look at how the coefficients work together, in this situation $x$ behaves only as a placeholder. So sometimes there are no solutions at all, we only use the properties of the polynomial.

• The input variable $x$ of a generating function (your example) is a "formal" variable in the sense that its value isn't really related to the problem we're trying to solve; but it usually seems to be treated as a real number, since we often take the derivative or integral of the generating function. If the generating function has a finite number of non-zero terms, I would consider it a polynomial with solutions; we just might not care what those solutions are. – David K Dec 21 '17 at 12:51
• I just wanted to give an other perspective of polynomials, still I consider them as symbols which can possibly take a value but not necessary. – Vinyl_cape_jawa Dec 21 '17 at 12:54