The Elements of the Theory of Algebraic Numbers

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3) The number 2 is associated with the square of a prime of the first degree.
It will be observed that the norm of every prime ?r of k(i) is a power (first or second) of a rational prime and that the degree of TT is the exponent of this power.
Moreover, we notice that 2 is the only rational prime that is divisible by the square of a prime of k(i) ; for, if this were true of any other rational prime of the form 4^+1, we should have TT associated with TT', and hence a + bi = a bi, a + bi, b + ai
...or b ai, which give a = o, b = o, or a = =^=b, all of which are seen to be incompatible with p = a 2 + b 2 .
10. Factorization of a Rational Prime in k(i) determined by the value of (d/p).
The rational primes may be classified with regard to their factorization in k(i) in the following manner: 1) Those of which the discriminant is a quadratic residue are factorable into two conjugate primes in k(i), called primes of the first degree. For (d/p) = i implies p = ^n-\-i, since d = 4, and we have seen that all rational primes of this form are thus factorable in k(i).


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