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.

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.

8 Statement 1 | R is a splitting field of some polynomial over Q. Statement 2 | There is a field with 60 elements.

Answer: Both Statement 1 and Statement 2 are false.

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 $\mathbb{R}$ were a splitting field of some polynomial over $\mathbb{Q}$ , then $\mathbb{R}$ would be a finite extension of $\mathbb{Q}$ . However, $\mathbb{R}$ is an infinite extension of $\mathbb{Q}$ (it is uncountable, while any finite extension of $\mathbb{Q}$ is countable). Therefore, $\mathbb{R}$ cannot be a splitting field of any polynomial over $\mathbb{Q}$ .

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., $q = p^n$ 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 $2^2 \cdot 3 \cdot 5$ . 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.

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