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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.

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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 $\mathbb R [x]$.

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