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There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I have yet to see an example anywhere of such a polynomial. Is this polynomial hard/impossible to compute? I've read the definition in the standard literature, however, none of the texts/books/papers that I've seen provide any examples of this polynomial. Sigh...

Any sightings of those polynomials out there? I would appreciate some feedback on this. Thanks.

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My third Google hit for "Iwasawa polynomial" is the note in Jeff's answer... –  Qiaochu Yuan Jun 4 '11 at 23:41
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2 Answers

This is an answer to the OP's question in the comment to Jeff's answer; since the OP has not asked it as a separate question, I am putting an answer here.

The Iwasawa polynomial is by definition a polynomial in $\mathbb Z_p[T]$, which is of the form $p^{\mu}f(x)$, where $\mu \geq 0$ and $f$ is monic of degree $\lambda$, with all non-leading coefficients divisible by $p$.

Since the coefficients are $p$-adic numbers, they can't be specified precisely, and so it is natural to compute them modulo some prescribed power of $p$ (as PARI apparently does). The most important pieces of information in the polynomial, though, are the quantities $\mu$ and $\lambda$ (these are called the $\mu$-invariant and $\lambda$-invariant of the $\mathbb Z_p$-extension under investigation, and are referred to collectively as the Iwasawa invariants of the extension), and these can be read off from the polynomial mod $p^n$, as soon as this reduction is non-zero (i.e. as soon as $n > \mu$).

In practice, I imagine, this means that one has to compute the Iwasawa polynomial modulo higher and higher powers of $p$ until you get a non-zero answer. For a general $\mathbb Z_p$-extensions, there is no known a priori bound on the $\mu$-invariant, as far as I know, and so one doesn't know in advance how far one will have to compute.

However, I believe that Iwasawa conjectured that the $\mu$-invariant always vanishes for the cylcotomic $\mathbb Z_p$-extension, and this is a theorem of Ferrero and Washington in the case when the ground field is abelian over $\mathbb Q$. So, in practice, when studying the cyclotomic $\mathbb Z_p$-extension of the field, you should already be able to read off the Iwasawa invariants just by computing modulo $p$ (and if you can't, you have disproved Iwasawa's conjecture, which would probably be even more exciting), and you are guaranteed that this is the case for an abelian ground field, e.g. in the case of a quadratic extension of $\mathbb Q$ (which was a case referred to by the OP in a now deleted follow-up question).

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Here is a function written for Pari/GP which computes Iwasawa polynomials. See in particular the note.

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The PARI/GP function computes $F_n(x)$ the Iwasawa polynomial $F(x)$ modulo $p^{n-r+1}$. Can we recover $F(x)$? –  Krasnicki Jun 7 '11 at 5:08
@Krasnicki: Dear Krasnicki, Yes (essentially); see my answer. Regards, –  Matt E Jun 7 '11 at 13:05
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