It is clear to understand what prime elements will be in case of ring of integers and many other rings. However, I find it confusing in case of fields, more specifically field of rational numbers. So will prime elements in ring of integers still be prime elements in the field of rational numbers? What would be the example of prime elements in the field?
2026-04-13 06:16:50.1776061010
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What would be a prime element in the field of rational numbers?
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As mentioned by Ferra, the field of rational numbers has no prime elements.
However, it is true that every nonzero rational number has a unique decomposition into a product of prime integers with positive and negative exponents. In this sense, the "primes" of the field of rational numbers are exactly the primes in the ring of integers.
The same holds for the field of fractions of every UFD.
In the field of rational numbers (and in any other field) there are no primes because all nonzero elements are units!