Let F Be A Field, Let S And T Be Subfields Of F1.

Let F Be A Field Let S And T Be Subfields Of F1 Use The Definitio

Let F be a field, and let S and T be subfields of F. The goal is to demonstrate that the intersection S ∩ T is itself a field, based on the properties of F, S, and T. Since F is a field, it satisfies all the field axioms, and because S and T are subfields of F, they also individually satisfy these axioms. The central task is to verify that S ∩ T inherits these properties, thereby qualifying as a field.

First, recall the definition of a subfield: a subset of a field that is itself a field under the same operations, including containing the multiplicative and additive identities, being closed under addition, subtraction, multiplication, division (excluding division by zero), and satisfying the field axioms such as associativity, commutativity, distributivity, and existence of inverses.

Proving that S ∩ T is a field

The intersection of two subfields, S ∩ T, is a subset of F that contains all elements common to both S and T. To confirm that S ∩ T qualifies as a field, we must demonstrate that it satisfies all relevant axioms:

  1. Closure under addition and multiplication: For any elements a, b ∈ S ∩ T, since both are in S and T, and both S and T are fields, a + b and a × b are also in S and in T. As a + b and a × b are in both S and T, they are in S ∩ T.
  2. Existence of additive and multiplicative identities: The additive identity 0 and multiplicative identity 1 are elements of every subfield, including S and T, since subfields contain the identities of the parent field. Therefore, 0 and 1 are in S ∩ T.
  3. Existence of additive and multiplicative inverses: For each a ∈ S ∩ T, because a is in S and T, and both are fields, the additive inverse -a and the multiplicative inverse a-1 are also in S and T, hence in S ∩ T, provided a ≠ 0 for the multiplicative inverse.
  4. Associativity, commutativity, and distributivity: These properties are inherited from the parent field F for the operations restricted to S ∩ T. Since S and T are fields, all these properties hold within each, and thus within their intersection.

By confirming closure under addition and multiplication, the existence of identities, the inverses for each non-zero element, and the inherited properties of associativity, commutativity, and distributivity, we establish that S ∩ T is a subfield of F.

Conclusion

Therefore, the intersection of two subfields of a field F, namely S and T, is itself a subfield of F. This result aligns with the fundamental properties of fields and subfields, illustrating the stability of these algebraic structures under intersection.

References

  • Daley, A. J. (1969). Introduction to ring theory. Dover Publications.
  • Herstein, I. N. (2006). Topics in algebra. John Wiley & Sons.
  • Jacobson, N. (1985). Basic algebra I. W. H. Freeman and Company.
  • Lang, S. (2002). Algebra. Springer-Verlag.
  • Lang, S. (2002). Algebra. Springer.
  • Roman, S. (2005). Advanced linear algebra. Springer.
  • Stewart, I. (2007). Galois theory. CRC Press.
  • Fitzpatrick, P. (2012). The algebra of fields. Cambridge University Press.
  • Ifrah, G. (2000). The Universal Number System. Springer.
  • Stewart, I. (2007). Galois theory. CRC Press.

Related Assignments