Proving Field Properties of R if F is Algebraic over K

In summary, if F is algebraic over K and R is a ring such that K is a subset of R which is a subset of F, then R is also a field. This can be shown by proving that every nonzero element in R has an inverse, which can be done by using either Hint A or Hint B: K(a) = K[a] and writing down the inverse of a in F, respectively.
  • #1
demonelite123
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Let [itex] K \subseteq F [/itex] be fields and let R be a ring such that [itex] K \subseteq R \subseteq F [/itex]. If F is algebraic over K, show that R is a field.

If F is algebraic over K, then every element of F is a root of some polynomial over K[x]. But since K is contained in R, every element of F is thus a root of some polynomial over R[x]. I want to show that every nonzero element of R has an inverse which would show that it is a field. The elements in K are obviously invertible so I need to show that any element in R that is not in K is also invertible. I am having trouble with this part and can't think of a way to show this. Can someone offer a hint or two in the right direction?

Help is greatly appreciated.
 
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  • #2


Let a be a nonzero element of R. Here are two independent (but ultimately equivalent) hints to help you show that the inverse of a belongs to R.

Hint A: K(a)=K[a].

Hint B: Write down the inverse of a in F, using the fact that a is algebraic/K.
 

1. What are the field properties of R if F is algebraic over K?

The field properties of R if F is algebraic over K are that R is a field, meaning it is closed under addition, multiplication, and inverses. It is also commutative, associative, and distributive. Additionally, R is algebraically closed, meaning every polynomial equation with coefficients in R has a solution in R.

2. How do you prove that R is a field if F is algebraic over K?

To prove that R is a field, you must show that it satisfies the defining properties of a field. This includes showing that it is closed under addition, multiplication, and inverses. Additionally, you must demonstrate that R is commutative, associative, and distributive. To prove algebraic closure, you must show that every polynomial equation with coefficients in R has a solution in R.

3. Can you provide an example of R being algebraically closed?

One example of R being algebraically closed is the field of complex numbers. Every polynomial equation with complex coefficients has a solution in the complex numbers, making it algebraically closed. This can be proven using the Fundamental Theorem of Algebra.

4. What is the significance of proving field properties of R if F is algebraic over K?

Proving the field properties of R if F is algebraic over K is important in understanding the structure and properties of these fields. It also allows for the use of algebraic techniques to solve equations and problems in these fields.

5. What are some applications of proving field properties of R if F is algebraic over K?

Knowing the field properties of R can be useful in many areas of mathematics and science. It can be applied in abstract algebra, number theory, and cryptography. Additionally, it has applications in physics, engineering, and computer science.

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