2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K such that f = a²). Let K(√√f) = {a+b√√f, a,b € K}. (a+b√√f)+(c + d√ƒ) = (a + c) + (b+d)√√√f; (a+b√√ƒ)(c+d√√ƒ) = (ac + bdf) + (ad + bc) √√ƒ. (a) Compute the degree [K(√f): K]. (b) Let P = K[x] be the polynomial P = x² - f. Show that P is irreducible. Deduce that K[x]/(P) is a field. (c) Let o: K(√ƒ) → K[x]/(P) be the map o(a+b√f) = a + bx + (P), where a, b € K. Show that is an isomorphism of fields.
2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K such that f = a²). Let K(√√f) = {a+b√√f, a,b € K}. (a+b√√f)+(c + d√ƒ) = (a + c) + (b+d)√√√f; (a+b√√ƒ)(c+d√√ƒ) = (ac + bdf) + (ad + bc) √√ƒ. (a) Compute the degree [K(√f): K]. (b) Let P = K[x] be the polynomial P = x² - f. Show that P is irreducible. Deduce that K[x]/(P) is a field. (c) Let o: K(√ƒ) → K[x]/(P) be the map o(a+b√f) = a + bx + (P), where a, b € K. Show that is an isomorphism of fields.
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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![2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K
such that f = a²). Let
K(√√f) = {a+b√√f, a,b € K}.
(a+b√√f)+(c + d√ƒ) = (a + c) + (b+d)√√√f;
(a+b√√ƒ)(c+d√√ƒ) = (ac + bdf) + (ad + bc) √√ƒ.
(a) Compute the degree [K(√f): K].
(b) Let P = K[x] be the polynomial P = x² - f. Show that P is irreducible.
Deduce that K[x]/(P) is a field.
(c) Let o: K(√ƒ) → K[x]/(P) be the map
o(a+b√f) = a + bx + (P),
where a, b € K. Show that is an isomorphism of fields.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F725ee8ea-573b-4075-8ffe-8d3f36956d72%2F7d73dd08-59d8-4b5e-a710-c7fca406b1e1%2Flcrz0j_processed.png&w=3840&q=75)
Transcribed Image Text:2. Let K be a field and let ƒ € K be a nonsquare (i.e. there is no element a € K
such that f = a²). Let
K(√√f) = {a+b√√f, a,b € K}.
(a+b√√f)+(c + d√ƒ) = (a + c) + (b+d)√√√f;
(a+b√√ƒ)(c+d√√ƒ) = (ac + bdf) + (ad + bc) √√ƒ.
(a) Compute the degree [K(√f): K].
(b) Let P = K[x] be the polynomial P = x² - f. Show that P is irreducible.
Deduce that K[x]/(P) is a field.
(c) Let o: K(√ƒ) → K[x]/(P) be the map
o(a+b√f) = a + bx + (P),
where a, b € K. Show that is an isomorphism of fields.
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