user30719: 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?

user30084: Elliptic curves are of the form y^2 = f(x).
This means there are two roots to the equation. i.e. if we know an x there are two possible y values that satisfy the equation (y & -y). Because we are using a modulo type number field, it happens that the even and odd translate to y and -y
https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus
This confused me too, until I figured out modulo type fields are not as normal as we expect.

user208: 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.

30719: <p>This is a paragraph from Andreas' book on compressed/uncompressed public keys. </p>

<p><em>Chapter 4 - Section - Key Formats</em> </p>

<blockquote>
  <p>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. <strong>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.</strong>
  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</p>
</blockquote>

<p>What I don't get is that bold text. </p>

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

<p>For example, are all even public keys below the x-axis?</p>
 30084: <p>Elliptic curves are of the form y^2 = f(x).</p>

<p>This means there are two roots to the equation. i.e. if we know an x there are two possible y values that satisfy the equation (y &amp; -y). Because we are using a modulo type number field, it happens that the even and odd translate to y and -y</p>

<p><a href="https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus" rel="nofollow">https://en.wikipedia.org/wiki/Quadratic_residue#Prime_or_prime_power_modulus</a></p>

<p>This confused me too, until I figured out modulo type fields are not as normal as we expect.</p>
 208: <p>There is no such thing as a negative or a positive value when you're talking in a finite field.</p>

<p>For example, in <code>Z7</code>, the field of integers modulo <code>7</code>. There holds:</p>

<ul>
<li><code>0</code> = <code>7</code> = <code>14</code> = <code>-7</code></li>
<li><code>1</code> = <code>8</code> = <code>15</code> = <code>-6</code></li>
<li><code>2</code> = <code>9</code> = <code>16</code> = <code>-5</code></li>
<li>...</li>
</ul>

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

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

<p>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.</p>
