4. Show that 7 is irreducible in the ring Z[V5] using the norm N defined by N(a + bv5) = | a? – 5b²|-

Q: Question 10

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Q: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 forall n >= 1.

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Q: Show that 7 is irreducible in the ring Z[V5].

A: Consider the ring ℤ5. Any element of the above ring is of the form a+b5 where a and b are integers.…

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

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Q: Rp { -b a | a, b = Zp}

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Q: с. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

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Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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Q: Let K be the splitting field of – 5 over Q. - • (a) Show that K = Q(V5,i/3) • (b) Explicitly…

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Q: 8. List all ideals of the ring Z12-

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Q: Use the field norm to show: a) 1+ 2 is a unit in Z [ 2]

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Q: C. Prove that neither 2 nor 17 are prime elements in Z[i] (the ring of Gaussian integers).

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Q: The ring Zpg?, has exactly-------------maximal ideals O 2

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