# Polynomials with $S_{p}$ as Galois group over $\mathbb{Q}$?

I have read the next beautiful result about Galois group of polynomials.

Theorem: Let $$f$$ be an irreducible polynomial of prime degree $$p\geqslant 5$$ in $$\mathbb{Q}[x]$$. If $$f$$ has exactly two nonreal roots, then the Galois group $$G_{f}=S_{p}$$.

I want to know whether this theorem gives a necessary and sufficient condition? Can we replace the condition in it with some others to get the same result?

Of course no. For example

$$\textbf{Proposition}.$$ There exists a polynomial $$R\in \mathbb{Q}[x]$$ of degree $$p$$, with only one real root, satisfying $$gal(R)=S_p$$.

$$\textbf{Proof}$$. We use the fact that the polynomials $$Q$$ with $$gal(Q)=S_p$$ are dense in $$\mathbb{Q}[x]$$.

Consider a monic polynomial $$T=x^p+\sum_{i with only one real root; randomly choose $$(b_i)$$ in a neighborhood of the $$(a_i)$$. Then, with probability $$1$$, $$T_1=x^p+\sum_{i has $$S_p$$ as Galois group. Moreover, $$T_1$$ has only one real root (by the continuity of the roots of a polynomial wrt. its coefficients, the non-real roots remain non-real).

Example. Let $$T=x(x^2+1)^2=x^5+2x^3+x$$ and

$$T_1=x^5+(2022376873266539/1000000000000000)x^3+(1040578712917309/1000000000000000)x+1141656374703/100000000000000+(97561097757381/1000000000000000)x^2+(77370700889713/1000000000000000)x^4\approx$$

$$x^5+2.022376873x^3+1.040578713x+0.01141656375+0.09756109776x^2+0.07737070089x^4$$.

• Do you have a reference for the stated fact? – Wojowu Mar 8 '19 at 9:46

This is false already for $$p=2$$, just take any irreducible quadratic polynomial with two real roots.

If you dislike even primes, try some cubic polynomials - recall that for an irreducible cubic $$f$$, we have either $$G_f=A_3$$ or $$S_3$$, and the former happens iff the discriminant is a square.

• I have edited that $p\geqslant 5$. – user450201 Mar 8 '19 at 10:01