# Introduction to Combinatorics (Chapman and Hall Mathematics by Alan Slomson

By Alan Slomson

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18. 19. Let k be any field, n a positive integer, and let a0 , . . , an−1 ∈ k. Compute x(a0 + a1 x + · · · + an−1 xn−1 ) in Rn . 3. 20. Let k be a field. A vector space V over k is an abelian group which admits a scalar multiplication by elements of k. If we let + denote the group operation and · (or concatenation) denote the scalar multiplication, then the following properties must be satisfied for any v, w ∈ V and any α, β ∈ k: • α(v + w) = αv + αw • (αβ)v = α(βv) • (α + β)v = αv + βv • 1k · v = v, where 1k is the multiplicative identity of k Elements of V are called vectors.

Let h(x) = x3 + x2 + 1 ∈ F2 [x]. a) Show that h(x) is irreducible in F2 [x]. Conclude that F := F2 [x]/ h(x) is a field. b) How many elements does F have? List them. Make an addition table and a multiplication table. 6 above. 8. Let m be a positive integer. Then there is a field with exactly m elements if and only if m = pn for some prime p and some positive integer n. Further, up to isomorphism, there is only one field with exactly pn elements, and it is of the form Fp [x]/ f (x) for some irreducible polynomial f (x) ∈ Fp [x] of degree n.

Then G can be written as a direct sum of cyclic groups. In fact, there are two canonical ways of doing this: • There are primes p1 , . . , pk and positive integers n1 , . . , nk such that G∼ = C n1 ⊕ · · · ⊕ C nk p1 pk • There are integers r1 , . . 2. 3. A ring is a set R equipped with two operations, usually denoted by + and · (or concatenation). 2. Rings, Fields, Ideals, and Factor Rings 47 ring, but we tend to call + addition and · multiplication in general. Every ring must satisfy all of the following properties: • Existence of Additive Identity: There is an element 0 ∈ R such that 0 + a = a = a + 0 for all a ∈ R.

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