When divisors go bad… counterexamples with polynomial division
... at most n roots. This result is false in R[x] when R has zero divisors. One example is f (x) = x2 − 1 in Z8 [x]. Before Test 1, we saw that all odd x ∈ Z satisfy x2 ≡ 1 (mod 8), so that means 1, 3, 5, 7 are four distinct roots of f (x) in Z8 . However, f only has degree 2. Another example is f (x) = ...
... at most n roots. This result is false in R[x] when R has zero divisors. One example is f (x) = x2 − 1 in Z8 [x]. Before Test 1, we saw that all odd x ∈ Z satisfy x2 ≡ 1 (mod 8), so that means 1, 3, 5, 7 are four distinct roots of f (x) in Z8 . However, f only has degree 2. Another example is f (x) = ...
1 Principal Ideal Domains
... of course ±6 don’t divide 2 + 2 −5 and so can’t be the GCD. We saw before that every Euclidean Domain R has the property that each ideal is generated by a single element. Let’s give this property a name. Definition. An integral domain in which every ideal is principal is called a Principal Ideal D ...
... of course ±6 don’t divide 2 + 2 −5 and so can’t be the GCD. We saw before that every Euclidean Domain R has the property that each ideal is generated by a single element. Let’s give this property a name. Definition. An integral domain in which every ideal is principal is called a Principal Ideal D ...
Summary for Chapter 5
... (Simplified form means: no powers of powers; each base appears only once; all fractions are in simplest form; no negative exponents) Simplify any polynomial by combining like terms Given a polynomial determine its degree and leading coefficient (remember terms are not always given in order from ...
... (Simplified form means: no powers of powers; each base appears only once; all fractions are in simplest form; no negative exponents) Simplify any polynomial by combining like terms Given a polynomial determine its degree and leading coefficient (remember terms are not always given in order from ...