You say you know that the "algebraic number form a field" and the field axioms but do you know the definition of "algebraic number". If x is an algebraic number, then there exist in integer, n, and integers, , , ..., and such that . Okay, what do you get if you divide each term in that equation by ?Could someone please give me a sketch of proof (or the proof itself) for the following claim. I'd appreciate it.
The claim is:
If is algebraic number prove that also is algebraic number.
Now I know a little bit of higher math/algebra and I know that algebraic numbers form a field and thus a quotient of two algebraic numbers is algebraic number. Is there a proof that does not require the axioms of field? Can someone give me an advice (or two)?
Thx in advance.
Or, since you know "a quotient of two algebraic numbers is algebraic number", 1/x is the quotient of 1 and x. Do you know that 1 is an algebraic number.
By the way, "a quotient of two algebraic numbers is algebraic number" is NOT strictly true. The denominator has to be non-zero. In fact, the original claim "If "x" is algebraic number then 1/x also is algebraic number" is NOT true. A counter example is x= 0. What is true is "If "x" is a non-zero algebraic number, then 1/x is also an algebraic number".