# Public points in SIDH

I was going through a presentation titled "SIKE in Hardware" by professor Reza Azarderakhsh. On the page $$10$$ of the presentation he introduces a variable $$w$$. Could you please explain what the variable is in layman's terms? I am a master student with a background in software engineering and little to none background in algebra.

Coordinates of points on the curves in SIDH are in a finite field $$\mathbf F_{p^2}$$.
First, $$\mathbf F_p$$ are integers mod $$p$$. Some polynomials on this field does not have roots, and what we can do about it is extend the field so it has a root. The same way $$X^2 + 1$$ does not have a root in real numbers, so we introduce $$i$$.
On this case, let's say we have an irreducible (it has no roots) polynomial of degree $$2$$ in $$\mathbf F_p$$, so we introduce a root $$w$$ and that's how we can create $$\mathbf F_{p^2}$$. Then, any element $$z$$ in this new field can be written $$z = a + bw$$ for $$a$$ and $$b$$ in $$\mathbf F_p$$.
Again, think of the complex numbers: anyone of them can be written $$a + bi$$ for $$a$$ and $$b$$ two real numbers.