0

I am looking for a way to factorize $x^{38} - 1$ polynomial in $F_5$ . So far I have: $x^{38} - 1 = (x^{19} - 1)(x^{19} + 1) = (x - 1)(x^{18} + x^{17}... + 1)(x + 1)(x^{18} - x^{17} + ... + 1)$

I can show that $(x^{18} + ... + 1)$ is irreducible due to minimal polynomial of primitive root of 19 in $F_5$ having degree 18(precisely $x^{18}+...+1$).

But I have no idea what to do with $x^{18} - x^{17}... +1$. I am not sure if it is irreducible or not.

Heisenbruh 2
  • 1
  • 1
  • $19\mid 5^9-1$ so the 19th root of unite resides in $\Bbb{F}{5^9}$, and hence it's minimal polynomial over $\Bbb{F}_5$ has degree nine only. But once you can factor the cyclotomic polynomial $\Phi{19}(x)$, the other factor is simply $\Phi_{19}(-x)$, and a factorization follows. You can use the law of quadratic reciprocity to conclude that $5$ is a quadratic residue modulo $19$, so degree $9$ is the worst you can expect. – Jyrki Lahtonen May 23 '23 at 10:26
  • 1
    See this old answer of mine for a bit of theory. – Jyrki Lahtonen May 23 '23 at 10:27
  • 1
    A "simpler" case factored. I use scare quotes, because there may be a trick that simplifies the process. A calculation similar to that in the linked thread shows that the constant terms of each of the monic degree nine factors of $\Phi_{19}(x)$ are both equal to $-1$. Another trick may say more... – Jyrki Lahtonen May 23 '23 at 10:40

1 Answers1

-1

The factor $x^{18}-x^{17}$ etc. is indeed irreducible. It corresponds to the numbers with a period of $38$.

DroneBetter
  • 145
wendy.krieger
  • 6,856
  • Also sprach Mathematica: $$x^{38}-1=(1 + x) (4 + x) (4 + 4 x + 2 x^2 + 4 x^3 + 2 x^4 + 2 x^5 + 2 x^6 + 3 x^7 + x^9) (1 + 4 x + 3 x^2 + 4 x^3 + 3 x^4 + 2 x^5 + 3 x^6 + 3 x^7 + x^9) (4 + 2 x^2 + 3 x^3 + 3 x^4 + 3 x^5 + x^6 + 3 x^7 + x^8 + x^9) (1 + 3 x^2 + 3 x^3 + 2 x^4 + 3 x^5 + 4 x^6 + 3 x^7 + 4 x^8 + x^9)$$ in $\Bbb{F}_5[x]$. See my comment under main as to why it has to factor further. – Jyrki Lahtonen May 23 '23 at 10:32
  • Factor[x^38-1,Modulus->5] if you want to test it. – Jyrki Lahtonen May 23 '23 at 10:33
  • So why not use -x? – wendy.krieger May 23 '23 at 10:35
  • PARI/GP can do this as well with "factormod(x^38-1,5)" – Peter May 23 '23 at 10:36
  • Mathematica writes $-x$ as $4x$, when I tell it to treat polynomials as if they have coefficients in the ring $\Bbb{Z}_5$. – Jyrki Lahtonen May 23 '23 at 10:37