On public keys compression, why an even or odd y coordinate corresponds to the positive / negative sign respectively? This is a paragraph from Andreas' book on compressed/uncompressed public keys. 

Chapter 4 - Section - Key Formats 


  Whereas uncompressed public keys have a prefix of 04 ,compressed
  public keys start with either a 02 or a 03 prefix. Let’s look at why
  there are two possible prefixes: because the left side of the equation
  is y^2 , that means the solution for y is a square root, which can
  have a positive or negative value. Visually, this means that the
  resulting y coordinate can be above the x-axis or below the x-axis. As
  you can see from the graph of the elliptic curve in Figure 4-2, the
  curve is symmetric, meaning it is reflected like a mirror by the
  x-axis. So, while we can omit the y coordinate we have to store the
  sign of y (positive or negative), or in other words, we have to
  remember if it was above or below the x-axis because each of those
  options represents a different point and a different public key. When
  calculating the elliptic curve in binary arithmetic on the finite
  field of prime order p, the y coordinate is either even or odd, which
  corresponds to the positive/negative sign as explained earlier.
  Therefore, to distinguish between the two possible values of y, we
  store a compressed public key with the prefix 02 if the y is even, and
  03 if it is odd, allowing the software to correctly deduce the y
  coordinate from the x coordinate and uncompress the public key to the
  full coordinates of the point. Public key compression is illustrated
  in Figure 4-7


What I don't get is that bold text. 

Why an even or odd y coordinate corresponds to the positive / negative sign?

For example, are all even public keys below the x-axis?
There is no such thing as a negative or a positive value when you're talking in a finite field.

For example, in Z7, the field of integers modulo 7. There holds:


0 = 7 = 14 = -7
1 = 8 = 15 = -6
2 = 9 = 16 = -5
...


So you can't say that the number 2 is positive, because it's equal to -5.

Despite that, the square root still has two solutions. For example, 3^2 = 9 = 2, 4^2 = 16 = 2. Thus both 3 and 4 are square roots of 2.

So we need a way to say which solution we want. Turns out, that when reduced to a range of 0-6, the two solutions of the square, one is odd and the other is even.