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15.2 Let F be an ordered field, and let a > 0. Show that if a has a square root in F, i.e., an element be F such that b² = a, then a has exactly two square roots in F, one of which is positive and the other negative. We use the notation √a to denote the posi- tive square root.

15.2 Let F be an ordered field, and let a > 0. Show that if a has a square root in F, i.e., an element be F such that b² = a, then a has exactly two square roots in F, one of which is positive and the other negative. We use the notation √a to denote the posi- tive square root.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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[Classical Geometries] How do you solve this question? Thank you

Definition An ordered field is a field F, together with a subset P, whose elements are called positive, satisfying: (i) If a, b e P, then a + b e P and ab € P. (ii) For any a € F, one and only one of the following holds: a e P; a = 0; −a = P. Here are a few elementary properties of an ordered field. Proposition 15.1 Let F, P be an ordered field. Then: (a) 1 € P, i.e., 1 is a positive element. (b) F has characteristic 0. (c) The smallest subfield of F containing 1 is isomorphic to the rational numbers Q. (d) For any a #0 € F, a² € P.
Transcribed Image Text:Definition An ordered field is a field F, together with a subset P, whose elements are called positive, satisfying: (i) If a, b e P, then a + b e P and ab € P. (ii) For any a € F, one and only one of the following holds: a e P; a = 0; −a = P. Here are a few elementary properties of an ordered field. Proposition 15.1 Let F, P be an ordered field. Then: (a) 1 € P, i.e., 1 is a positive element. (b) F has characteristic 0. (c) The smallest subfield of F containing 1 is isomorphic to the rational numbers Q. (d) For any a #0 € F, a² € P.
15.2 Let F be an ordered field, and let a > 0. Show that if a has a square root in F, i.e., an element be F such that b² = a, then a has exactly two square roots in F, one of which is positive and the other negative. We use the notation √a to denote the posi- tive square root.
Transcribed Image Text:15.2 Let F be an ordered field, and let a > 0. Show that if a has a square root in F, i.e., an element be F such that b² = a, then a has exactly two square roots in F, one of which is positive and the other negative. We use the notation √a to denote the posi- tive square root.
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