THis is really long but wanted to check my work

FInd all eighth-roots of 1. Use this to express f (x)=x^8-1 as a product of irreducible polynomials

a) in C [x]

b) in R [x]

c) in Q [x]

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- April 8th 2007, 07:34 PMchadlyterIrreducible polys
THis is really long but wanted to check my work

FInd all eighth-roots of 1. Use this to express f (x)=x^8-1 as a product of irreducible polynomials

a) in C [x]

b) in R [x]

c) in Q [x] - April 9th 2007, 05:48 AMCaptainBlack
- April 9th 2007, 08:32 AMSoroban
Hello, chadlyter!

Quote:

Find all eighth-roots of 1.

Use this to express f (x) = x^8 -1 as a product of irreducible polynomials

a) in C[x] . . b) in R[x] . . c) in Q[x]

(c) Factor the polynomial.

x^8 - 1 .= .(x^4 - 1)(x^4 + 1) .= .(x² - 1)(x² + 1)(x^4 + 1)

. . = .(x - 1)(x + 1)(x² + 1)(x^4 + 1)

(b) The quartic factor can be factored:

. . . . . . . . . . . . . . . . . . . . . . - . . - . ._ . . . . . . . . . _

x^8 - 1 .= .(x - 1)(x + 1)(x² + 1)(x² - √2x + 1)(x² + √2x + 1)

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ._

(a) The eight roots are: . ±1, ±*i*, (±1 ±*i*)/√2

To save me typing all those ugly radicals,

. . let: .x1 .= .(1 +*i*)/√2, .x2 .= .(1 -*i*)/√2, .x3 .= .(-1 +*i*)/√2, .x4 .= .(-1 -*i*)/√2

x^8 - 1 .= .(x - 1)(x + 1)(x -*i*)(x +*i*)(x - x1)(x - x2)(x - x3)(x - x4)