document.write( "Question 1123424: find the gcd of 3+4i and 4+3i in the ring Z[i] \n" ); document.write( "

Algebra.Com's Answer #739735 by rothauserc(4718) $\"\"$ $\"About$

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look at the norms
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\n" ); document.write( "N(3+4i) = 9 + 16 = 25
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\n" ); document.write( "N(4+3i) = 16 + 9 = 25
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\n" ); document.write( "the primes dividing each have to divide the rational prime 5, since 5^2 = 25
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\n" ); document.write( "these are (2+i), (2-i)
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\n" ); document.write( "check if (2+i) divides (3+4i)
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\n" ); document.write( "(3+4i)/(2+i) = (3+4i)(2-i)/(2+i)(2-i) = (10+5i)/5
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\n" ); document.write( "so (2+i) divides (3+4i)
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\n" ); document.write( "now check if (2+i) divides (4+3i)
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\n" ); document.write( "(4+3i)/(2+i) = (4+3i)(2-i)/(2+i)(2-i) = (11+2i)/5
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\n" ); document.write( "(2+i) does not divide (4+3i) and it follow that (2-i) divides (4+3i)
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\n" ); document.write( "check if (2-i) divides (3+4i)
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\n" ); document.write( "(3+4i)/(2-i) = (3+4i)(2+i)/(2-i)(2+i) = (2+8i)/5
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\n" ); document.write( "(2-i) does not divide (3+4i)
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\n" ); document.write( "therefore we can conclude that the gcd is 1
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