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Exercise 4

Avg difficulty 3.0

Let FF be a field and nn be a non-negative integer. Prove that

V={f(x)∈F[x]∣deg⁡f(x)≤n}∪{0}V=\{f(x)\in F[x]\mid\deg f(x)\leq n\}\cup\{0\}

is a vector space over FF using the definitions of polynomial addition and scalar multiplication.(Warning: be careful with the zero polynomial!)

Kan, A Complete Introduction to Abstract Algebra, Fields, Vector Spaces, Exercise 4

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Adapted from A Complete Introduction to Abstract Algebra © 2022–2026 Kan Onn Kit. Licensed under CC BY-SA 4.0. Source: A Complete Introduction to Abstract Algebra. Imported at revision a559e8b.