Surprisingly, I understood some of that (not a strong math background), but I asked Bing AI to explain it at the level a smart high schooler would get it. How did it do?:

Let me try to restate this in a simpler way. A polynomial is a mathematical expression that involves adding and multiplying powers of a variable, such as x. For example, x^2 + 3x - 5 is a polynomial of degree 2. A coefficient is the number that multiplies a power of x, such as 3 in 3x. A coefficient vector is a list of all the coefficients of a polynomial in order from highest to lowest power of x, such as [1, 3, -5] for x^2 + 3x - 5.

Sometimes, we want to know what happens when we plug in different values of x into a polynomial. For example, what is the value of x^2 + 3x - 5 when x = 2? This is called evaluating the polynomial at a point. We can also evaluate the polynomial at many points at once, such as x = 1, 2, 3, 4. This is called evaluating the polynomial at a set of points.

The Fourier transform is a technique that helps us do this faster and easier. It takes a coefficient vector of a polynomial and transforms it into another vector that contains the values of the polynomial at a special set of points. These points are called powers of a primitive Nth root of unity. This sounds complicated, but it just means that they are numbers that when multiplied by themselves N times give 1. For example, when N = 4, one such number is i (the imaginary unit), because i^4 = 1. The powers of i are i^0 = 1, i^1 = i, i^2 = -1, i^3 = -i. These are the four points that we can evaluate any polynomial of degree 3 or less using the Fourier transform.

The inverse Fourier transform does the opposite. It takes a vector that contains the values of a polynomial at some points and transforms it back into the coefficient vector of the polynomial. This is useful when we want to find the polynomial that passes through some given points. This is called interpolating the polynomial from the points.

The Fourier transform and the inverse Fourier transform are very similar, except that they use different orders of the coefficients and the points, and they have to divide or multiply by some number (the scaling factor) to make sure everything works out.

The nice thing about finite fields is that they always have some numbers that can be used as powers of a primitive Nth root of unity for any N. A finite field is a set of numbers that has a fixed size and follows certain rules for adding and multiplying. For example, GF(7) is a finite field with 7 elements: 0, 1, 2, 3, 4, 5, 6. In GF(7), we add and multiply by taking the remainder after dividing by 7. For example, 4 + 5 = 9 mod 7 = 2 in GF(7). In GF(7), one primitive Nth root of unity for any N is 3, because 3^N mod 7 = 1 for any N that is divisible by 6. The powers of 3 are 3^0 = 1, 3^1 = 3, 3^2 = 9 mod 7 = 2, ..., and so on.

This means that we can use the Fourier transform and the inverse Fourier transform to evaluate and interpolate any polynomial in any finite field without worrying about finding the right points to use.

I hope this helps you understand better what the original text was saying. If you have any questions or want more examples, feel free to ask me.