FOURTH PROBLEM SHEET FOR ALGEBRAIC NUMBER THEORY
... 4. Show that associating to an ideal I ⊆ OK its conjugate ideal I preserves congruence classes of ideals modulo the group of principal ideals, so that conjugation is a well-defined operation on Cl(K). Prove that an element of Cl(K) is fixed by conjugation if and only if it has order at most 2 (that ...
... 4. Show that associating to an ideal I ⊆ OK its conjugate ideal I preserves congruence classes of ideals modulo the group of principal ideals, so that conjugation is a well-defined operation on Cl(K). Prove that an element of Cl(K) is fixed by conjugation if and only if it has order at most 2 (that ...
solutions for the practice test
... So the boundary line is y = 13 x + 2. The boundary line does not satisfy the inequality, because it is “>” and not “≥”. Finally, we want the half-plane above the line, since y is greater than the line. This give us the following picture: ...
... So the boundary line is y = 13 x + 2. The boundary line does not satisfy the inequality, because it is “>” and not “≥”. Finally, we want the half-plane above the line, since y is greater than the line. This give us the following picture: ...
MA2215: Fields, rings, and modules
... 1. Clearly, it is enough to check it for f(x) = xk , since every polynomial is a linear combination of these, and if x − a divides each of the summands, it divides the whole sum too. But xk − ak = (x − a)(xk−1 + xk−2 a + . . . + xak−2 + ak−1 ). The statement about the roots is clear: f(x) = q(x)(x − ...
... 1. Clearly, it is enough to check it for f(x) = xk , since every polynomial is a linear combination of these, and if x − a divides each of the summands, it divides the whole sum too. But xk − ak = (x − a)(xk−1 + xk−2 a + . . . + xak−2 + ak−1 ). The statement about the roots is clear: f(x) = q(x)(x − ...
CHAP12 Polynomial Codes
... are coded for reasons of secrecy. But there's another reason why we might want to code a message. We might want to transmit across a “noisy” channel, that is, a channel that could corrupt the text. Of course we could simply repeat the message. Wherever the two copies differ, the receiver will know t ...
... are coded for reasons of secrecy. But there's another reason why we might want to code a message. We might want to transmit across a “noisy” channel, that is, a channel that could corrupt the text. Of course we could simply repeat the message. Wherever the two copies differ, the receiver will know t ...