Question

6. Show that (i) x^4 + 1 is irreducible over Z. (ii) x^5 + 15x^4 + 20x^2 + 21x + 1 is irreducible over Z. (iii) x^4 − x^3 + x^2 − x + 1 is irreducible over Q. (iv) x^4 − 5x^2 + x + 1 is irreducible over Q. (v) 2x^7 + 10x^5 + 25x^4 + 5x + 10 is irreducible over Q.

Name: 6 show that i x4 1 is irreducible over z ii x5 15x4 20x2 21x 1 is irreducible over z iii x4 x3 x2 x 1 is irreducible over q iv x4 5x2 x 1 is irreducible over q v 2x7 10x5 25x4 5x 10 is irred 80425
Uploaded: 2023-06-21T05:09:13-08:00
Duration: 2 min 41 s
Channel: Adi S
Description: 6 show that i x4 1 is irreducible over z ii x5 15x4 20x2 21x 1 is irreducible over z iii x4 x3 x2 x 1 is irreducible over q iv x4 5x2 x 1 is irreducible over q v 2x7 10x5 25x4 5x 10 is irred 80425

          6. Show that (i) x^4 + 1 is irreducible over Z.
(ii) x^5 + 15x^4 + 20x^2 + 21x + 1 is irreducible over Z.
(iii) x^4 − x^3 + x^2 − x + 1 is irreducible over Q.
(iv) x^4 − 5x^2 + x + 1 is irreducible over Q.
(v) 2x^7 + 10x^5 + 25x^4 + 5x + 10 is irreducible over Q.

Added by Rachel P.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

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Solved by Expert Adi S

06/21/2023

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According to the criterion, if there exists a prime number p such that: Show more…

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6. Show that (i) x^4 + 1 is irreducible over Z. (ii) x^5 + 15x^4 + 20x^2 + 21x + 1 is irreducible over Z. (iii) x^4 − x^3 + x^2 − x + 1 is irreducible over Q. (iv) x^4 − 5x^2 + x + 1 is irreducible over Q. (v) 2x^7 + 10x^5 + 25x^4 + 5x + 10 is irreducible over Q.