Visual Group Theory Lecture 72 Ideals quotient rings and finite fields

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Visual Group Theory, Lecture 7.2: Ideals, quotient rings, and finite fields • A left (resp., right) ideal of a ring R is a subring that is invariant under left (resp., right) multiplication. Two-sided ideals are those that are both left and right ideals. This is the analogue of normal subgroups, in that the quotient ring R/I is well-defined iff I is a two-sided ideal. If one takes the quotient of the polynomial ring over Z_p (the integers modulo p) by an irreducible polynomial of degree n, then the result is a finite field of order q=p^n. It turns out that up to isomorphism, there is a unique finite field of each order q=p^n. Thus, all finite fields are of the form Z_p for a prime p, or Z_p[x]/(f) for an irreducible polynomial f. • Course webpage (with lecture notes, HW, etc.): http://www.math.clemson.edu/~macaule/...

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