# Question on irreducibility of curves over $\mathbb{R}$

Suppose we look at the elliptic curve $y^2 = x^3 - x$ over $\mathbb{R}$. Then this has two connected components in the Euclidean topology because the cubic has three real roots. However, the polynomial $y^2 - x^3+x$ is irreducible in $\mathbb{R}[x,y]$, so it generates a prime ideal. Thus the vanishing set (over $\mathbb{R}$) should be irreducible as an algebraic set. Is this correct even though the curve "looks" (in the Euclidean sense, but perhaps not Zariski sense) like the union of two components?

• Have you thought of $x^2+1\in\mathbb{R}[x]$? It defines the empty set over reals or $x^2+y^2\in\mathbb{R}[x,y]$? Defines the origin over the reals, but not irreducible over complex numbers. Real points do not tell you enough about the underlying variety. Commented Apr 27, 2016 at 1:50
• So you are saying that having two connected components over $\mathbb{R}$ should be irrelevant to the "true" picture over the complex numbers? (Hence my example is still irreducible as a curve.) Commented Apr 27, 2016 at 2:09
• Not irrelevant, real points tell you something, as does rational points, important for a number theorist. But, irreducibility has a definition, which may not be apparent by looking at real or rational points. Commented Apr 27, 2016 at 2:12

When you visualize what $\text{Spec } \mathbb{R}[x, y]/(y^2 - x^3 + x)$ looks like to an algebraic geometer, you should not be visualizing its real points: rather, you should be visualizing its complex points, together with the action of complex conjugation on them. The fixed points of complex conjugation are the real points, but there is a much richer set of complex points that aren't fixed, and it's these that are responsible for connecting together the two real connected components.
• Mohan's example of $\text{Spec } \mathbb{R}[x, y]/(x^2 + y^2)$ in the comments is more complicated; the base change to $\mathbb{C}$ is not irreducible, but it is. This reflects the fact that its complex points consist of two irreducible components which are sent to each other by complex conjugation. Commented Apr 27, 2016 at 5:37