1. In order to understand that Gaussian or Eisenstein integers have a unique decomposition in terms of prime (G/E) integers, it has to be understood that the decomposition can differ only by __________________ and ____________. 2. For example, for the Gaussian integer 5, i.e., 5 + 0i, the decomposition
1. In order to understand that Gaussian or Eisenstein integers have a unique decomposition in terms of prime (G/E) integers, it has to be understood that the decomposition can differ only by __________________ and ____________.
2. For example, for the Gaussian integer 5, i.e., 5 + 0i, the decomposition 5=(1+2i)(1−2i) and the decomposition 5=(2+i)(2−i) look different, but their difference can be resolved by noticing that each factor in the expression (1+2i)(1−2i) is a rearrangement of the factors in the expression (2+i)(2−i) after multiplying by the appropriate unit. That is, (1+2i)=u⋅(2−i), where the unit u = _____ and (1−2i)=v⋅(2+i), where the unit v = _____.
Question 1
- The decomposition can differ only by signs and units.
Question 2
- The units u and v can be chosen to be 1+i and 1-i, respectively.
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