By Julian Lowell Coolidge

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Suppose d 1 and d 2 are positive numbers in field F but neither is a square in F. 19. Do IQ( v'2, y'5) and IQ( y'5, v'2) contain the same elements of lR? Do IQ( y'6, v'2) and IQ( y'6, v'3) contain the same elements of lR? 20. 14 have a converse that is true? 21. Show how Descartes used the parabola with equation y = x 2 and the circle through (0,0) with center (k/2, 1/2) to construct cube roots and used the parabola with equation y = x 2 and the circle through (0,0) with center (cos 3A, 2) to trisect angles.

The coordinate axes are ruler and compass lines. All of (p,O), (-p,O), (O,p), and (0, -p) are ruler and compaS8 points if anyone is a ruler and compass point. Number x is a ruler and compass number iff -x is a ruler and compass number. The integers are ruler and compass numbers. Point (p, q) is a ruler and compass point iff both p and q are ruler and compass numbers. 34 2. 2. Just because the X-axis is a ruler and compass line, it does not follow that every point on the X-axis is a ruler and compass point.

Likewise, not every point on a ruler and compass circle is a ruler and compass point. Recall that a rational number is a quotient min where m and n are integers with n #- O. A real number that is not a rational is called an irrational number. You may know a more general definition of the word "field" than that given below. However, such a generalization is not necessary for our purposes. 5. A field F is a subset of the real numbers that contains 0 and 1 and such that a+b, a-b, ab, alc are in F whenever a,b,c are in F but c #- o.