# Irreducible Polynomials over $\mathbb{R}$ or $\mathbb{Q}$

I am interested in generating irreducible polynomials of a given, arbitrary degree over either the reals or rationals using integer coefficients. They don't necessarily have to be arbitrary polynomials (i.e. random coefficients), but I would prefer not to simply use $F(x)=x^n+a_n$ (for the case of $\mathbb{R}$.

If a general algorithm for generating arbitrary irreducible polynomials does not exist, I am open to using several "families" of polynomials of arbitrary degree which are known to be irreducible, in which case I would like to be able to generate several different such families.

• There don't exist irreducible polynomials of arbitrary degree over $\mathbf{R}$: all polynomials factor into a product of linear and quadratic factors. – Hurkyl Apr 8 '14 at 16:35
• Irreducible polynomials over $\Bbb{R}$ are precisely the constants, the linear polynomials, and the quadratic polynomials with negative discriminant. – Servaes Apr 8 '14 at 16:36
• Hmm, I didn't realize that about the reals. Thanks! – Foo Barrigno Apr 8 '14 at 16:44
• You may enjoy looking into irreducible polynomials over finite fields; this is quite interesting. For example, consider polynomials whose coefficients are in $\Bbb Z_2$, which means the field containing only $0$ and $1$, with addition and multiplication as you expect, except that $1+1=0$. There is a unique irreducible polynomial of degree 2, namely $x^2+x+1$, and of 8 possible degree-3 polynomials, exactly 2 are irreducible. There is a lot of very fascinating theory at work here, including a lovely theorem of Gauss that counts of irreducible polynomials of each order. – MJD Apr 8 '14 at 17:01
• @MJD: I'm definitely interested in it, I'm just trying to narrow down certain results to rational (and I guess not real) polynomials for now. I plan on tackling some fun finite fields soon, too. In which case I've seen several results (such as results by Shoup) on constructing irreducibles in finite fields. – Foo Barrigno Apr 8 '14 at 17:05

Over the reals is not interesting, no polynomial of degree $\ge 3$ is irreducible.