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).
What to read after The Elements of the Theory of Algebraic Numbers?
You can find similar books in the "Read Also" column, or choose other free books by Legh Wilber Reid to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
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).
What to read after The Elements of the Theory of Algebraic Numbers?
You can find similar books in the "Read Also" column, or choose other free books by Legh Wilber Reid to read onlineMoreLess
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