7.25. This exercise asks you to prove Proposition 7.39, which says that permutations € S act on the ring of R[X₁,..., Xn] as homomorphisms. (Hint. The hardest part of this problem is figuring out a good notation for multivariate polynomial rings.) (a) Prove that sends sums to sums, π(p+q) = π(p) + n(q). (b) Prove that sends products to products, π(p-q) = π(p) - π(q). (c) Let σ € ST be another permutation. Prove that π (σ (p)) = (no)(p).
7.25. This exercise asks you to prove Proposition 7.39, which says that permutations € S act on the ring of R[X₁,..., Xn] as homomorphisms. (Hint. The hardest part of this problem is figuring out a good notation for multivariate polynomial rings.) (a) Prove that sends sums to sums, π(p+q) = π(p) + n(q). (b) Prove that sends products to products, π(p-q) = π(p) - π(q). (c) Let σ € ST be another permutation. Prove that π (σ (p)) = (no)(p).
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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![7.25. This exercise asks you to prove Proposition 7.39, which says that permutations € S₁, act on
the ring of R[X₁,..., Xn] as homomorphisms. (Hint. The hardest part of this problem is figuring
out a good notation for multivariate polynomial rings.)
sends sums to sums, (p+q) = π(p) + π(g).
(a) Prove that
(b) Prove that
sends products to products, (p. q) = (p). π(q).
(c) Let σ € S₁, be another permutation. Prove that (o (p)) = (no)(p).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc98c3d06-b4a0-47ab-9898-9b16a61ca414%2F24c2f99f-563a-4f79-8244-8046c9e3fd64%2Fd9l0j5_processed.png&w=3840&q=75)
Transcribed Image Text:7.25. This exercise asks you to prove Proposition 7.39, which says that permutations € S₁, act on
the ring of R[X₁,..., Xn] as homomorphisms. (Hint. The hardest part of this problem is figuring
out a good notation for multivariate polynomial rings.)
sends sums to sums, (p+q) = π(p) + π(g).
(a) Prove that
(b) Prove that
sends products to products, (p. q) = (p). π(q).
(c) Let σ € S₁, be another permutation. Prove that (o (p)) = (no)(p).
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