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I just asked wolframalpha to factor $X^{16}+X$ over $\mathbb{F}_2$. The normal factorization is $$ X(X+1)(X^2-X+1)(X^4-X^3+X^2-X+1)(X^8+X^7-X^5-X^4-X^3+X+1) $$ and over $GF(2)$ it is $$ X(X+1)(X^2+X+1)(X^4+X+1)(X^4+X^3+1)(X^4+X^3+X^2+X+1). $$ Does the second form follow from the first, or is there a different way to factor over $\mathbb{F}_2$? I noticed that simply replacing the $-$ signs with $+$ signs in the first factorization doesn't yield the second one.

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    $\begingroup$ The last term of the first is (mod $2$) the product of the $4$th and $5$th term of the second. The first $4$ terms of the first were irreducible mod $2$. $\endgroup$ – André Nicolas Jul 12 '11 at 7:16
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It shouldn't be a surprise that switching from integer (or rational) coefficients to modular coefficients allows for further factorization. The simplest example is probably the factorization $$ x^2+1=x^2+2x+1=(x+1)^2 $$ over $F_2$.

Here the key difference between the two factorizations is that the polynomial of degree 8 splits into a product of two quartic polynomials over $F_2$: $$(x^4+x+1)(x^4+x^3+1)=x^8+x^7+x^5+x^4+x^3+x+1$$ that is equal to (up to sign changes) your last factor.

Once you start on finite fields in your studies you will immediately learn that all the elements of $GF(16)$ are roots of the polynomial $x^{16}+x$. As that field is a degree 4 extension of $F_2$, all those elements have minimal polynomials of degree a factor of 4. Furthermore, all such irreducible polynomials appear as factors of $x^{16}+x$. Now, we easily see that both $x^4+x+1$ and its reciprocal polynomial $x^4+x^3+1$ are both irreducible in the ring $F_2[x]$, so they must appear as factors.

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  • $\begingroup$ In $\mathbb{F}_2[x]$, why can't we just factor $x^{16}-x=x(x+1)$? It seems to me that $x$ and $(x+1)$ should be irreducible since they have degree 1. $\endgroup$ – Pascal's Wager Feb 28 '18 at 20:12
  • $\begingroup$ @Pascal'sWager Yes, $x$ and $x+1$ are both irreducible. But their product has degree two whereas $x^{16}-x$ has degree $16$. In other words $x^{16}-x\neq x(x+1)$, Why did you think this would be a factorization? $\endgroup$ – Jyrki Lahtonen Feb 28 '18 at 20:17
  • $\begingroup$ I thought it would be a factorization since $x^{16}-x=x(x+1)$ for any $x \in \mathbb{F}_2$. $\endgroup$ – Pascal's Wager Feb 28 '18 at 20:31
  • $\begingroup$ @Pascal'sWager But we are dealing with formal polynomials here. Not polynomial functions. In other words, you should not think of $x$ as a variable ranging over $\Bbb{F}_2$. It is an indeterminate. You may want to study this thread. $\endgroup$ – Jyrki Lahtonen Feb 28 '18 at 20:37
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The second form does follow from the first. Note that over $\mathbb F_2$, the pair of polynomials $$ X^2 - X + 1 \quad \text{ and } \quad X^2 + X + 1 $$ and the other pair $$ X^4 - X^3 + X^2 - X + 1 \quad \text{ and } \quad X^4 + X^3 + X^2 + X + 1 $$ are the same. The only difference that comes up in $\mathbb F_2$ is that the polynomial of degree $8$ factors.

Hope that helps,

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