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Statement 1 | R is a splitting field of some polynomial over Q. Statement 2 | There is a field with 60 elements.
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Step-by-Step Solution
Step 1: Analyze Statement 1
A splitting field of a polynomial f(x) over a field F is the smallest field extension of F over which f(x) splits into linear factors. If R were a splitting field of some polynomial over Q, then R would be a finite extension of Q. However, R is an infinite extension of Q (it is uncountable, while any finite extension of Q is countable). Therefore, R cannot be a splitting field of any polynomial over Q.
Step 2: Analyze Statement 2
A fundamental theorem in field theory states that a finite field with q elements exists if and only if q is a prime power, i.e., q=pn for some prime number p and positive integer n. We need to check if 60 can be expressed in this form. The prime factorization of 60 is 22⋅3⋅5. Since 60 is not a prime power, there is no field with 60 elements.
Step 3: Conclusion
Based on the analysis of each statement, both Statement 1 and Statement 2 are false.