# Elements of the spectrum of complex numbers

I recently learned that the elements in the spectrum of $$\mathbb{C}[x]$$ are in the form $$x-a$$. I understand that a spectrum consists of all prime ideals of a ring, but I'm a little confused as to why for the complex numbers, this means that the elements of the spectrum are in the form mentioned above.

In addition, I understand that the elements of $$\mathbb{R}[x]$$ are also of this form, but also contain elements that irreducible quadratics.

I need some help understanding why this is the form of elements in both of these rings. Thank you!

Please keep in my mind that I have taken a ring theory class, but I haven't learned anything about topology.

Check out Ravi Vakil's notes

On page 102 he gives an elementary proof that the prime ideals of $$\mathbb C[x]$$ are either the zero ideal $$(0)$$ or of the form $$(x-a)$$, where $$a \in \mathbb C$$ using only that $$\mathbb C [x]$$ has a division algorithm, and is a unique factorization domain. Example 5 On page 104 he states what one gets if one proceeds similarly for $$\mathbb R [x]$$.