# 4. Let N be a finite group and let H be a subgroup of N. If |H| is ...

## Transcribed Text

4. Let N be a finite group and let H be a subgroup of N. If |H| is odd and [N:H] = 2, prove that the product of all of the elements of N, in any order, cannot belong to H. Solution Reflective Narrative 5. Prove that ℤ[𝑖]/〈1 − 𝑖〉 is a field. Solution Reflective Narrative 6. Suppose that 𝑓(𝑥) = 𝑥 𝑛 + 𝑎𝑛−1𝑥 𝑛−1 + ⋯ + 𝑎0 ∈ ℤ[𝑥]. If 𝑟 is rational and 𝑥 − 𝑟 divides 𝑓(𝑥), prove that 𝑟 is an integer. Solution Reflective Narrative 7. In the field GF(5n), where n is even, prove that 𝑥 5 𝑛 − 𝑥 has an irreducible factor over GF(5) of degree 2. Solution Reflective Narrative

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