Posted by: **Alexandre Borovik** | May 24, 2008

## Fermat’s Last Theorem for Polynomials

An old post by Nick Halloway at sci.math:

Someone mentioned to that an analog of FLT is true in as well, i.e. there are no relatively prime [non-constant] polynomials , and in with

for .

This is true. You can show it by adapting the false proof of FLT.

It’s only necessary to show it for an odd prime and for . The proof is similar for to an odd prime, so I will just show it for odd primes.

**Proof** by induction on the sum of the degrees of and .

Suppose the sum of the degrees is . Then factors as

,

where is a primitive ‘th root of unity. These factors are all relatively prime polynomials of degree . But has repeated factors. So it won’t work for this case.

Now assume the sum of the degrees is . Again factor as

These factors are relatively prime. But has factors with multiplicity divisible by . So each factor must be a ‘th power. So we have

,

,

So , and

So

Since neither nor are 0, and and are relatively prime, this violates the induction hypothesis.

This also shows it for polynomials in , since polynomials that are relatively prime in are also relatively prime in .

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The remark at the end is false. For example, consider the constant polynomials f=2 and g=3. These are relatively prime in Z[x], but are multiples of each other in C[x].

By:

Lucason May 24, 2008at 4:27 pm

Two elements f, g in a UFD are relatively prime if there is no irreducible or prime element which divides them both. Since 2 and 3 are units in C[x], neither has a prime factor, so they are relatively prime in this ring.

By:

Todd Trimbleon May 25, 2008at 12:52 pm

Yes, you are right (I wasn’t thinking clearly), but then the result is false. We have that 2, 3, and (2^3+3^3)^(1/3) are all units in the ring, and clearly these yield a solution of a^n+b^n=c^n, with n=3.

In point of fact, now that I look at it, the proof assumes that the polynomials are nonconstant, but that isn’t stated in the statement of the theorem.

By:

Lucason May 25, 2008at 3:14 pm