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B/ for all 2 ᏼ. an // is an lcm 2ᏼ of a1 ; : : : ; an (where 1 is to be understood as 0 if it occurs). Proof. Part (i) follows easily from F10 with the help of (28). x/ 0 for all , then x 2 R, by (29). The converse is clear, so (ii) is established. Since a j b is equivalent to b=a 2 R, part (iii) follows using (28). Part (iv) now is an automatic consequence of (iii). ˜ 5. The foregoing sections have dealt with little more than the general foundations of elementary arithmetic. 2. Definition 8. Let R be a (not necessarily commutative) ring with unity 1 ¤ 0.

Gauss’s Theorem 47 Given a 2 R, we consider in particular the quotient map R ! R=a and its natural extension (4) RŒX  ! R=a/ŒX : F4. (i) The homomorphism (4) yields a natural isomorphism (of R-algebras) RŒX =a ! R=a/ŒX : (ii) An element a 2 R is prime in R if and only if it is prime in RŒX . Proof. Part (i) follows from the Fundamental Homomorphism Theorem, since the kernel of (4) is clearly I D aRŒX . R=a/ŒX  is an integral domain ” RŒX =a is an integral domain ” a is prime in RŒX .

But you should keep an eye open in each case for whether the n represents an integer or an element of K. (b) Clearly ‫ ޑ‬is a prime field (indeed, up to isomorphism, the only prime field of characteristic 0). For any prime number p, ‫ކ‬p WD ‫= ޚ‬p ‫ޚ‬ (31) is a field (see Chapter 2, Remark after F2; naturally, to show that ‫= ޚ‬p ‫ ޚ‬has no zero-divisors, it is necessary to use the well-known Euclidean result: if p is a prime dividing ab, then p divides a or b; see also Chapter 4). For a given p, the field ‫ކ‬p is, up to isomorphism, the only prime field of characteristic p.

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