Abstract Algebra: Splitting Fields and Prime Polynomials

In summary, the conversation discusses the concept of splitting fields and their degree over the rational numbers. The degree of a splitting field can be determined by finding the smallest extension of the rational numbers that contains all its roots. In the case of x^4 + 1, the splitting field would be Q(\sqrt{2},i) with a degree of 4. The conversation also touches on the concept of algebraic numbers and their role in determining the degree of an extension field. Overall, the conversation provides helpful advice for understanding and solving problems related to splitting fields.
  • #1
buzzmath
112
0
I'm having trouble understanding splitting fields. Some of the problems are find the degree of the splitting field of x^4 + 1 over the rational numbers and if p is a prime prove that the splitting field over the rationals of the polynomial x^p - 1 is of degree p-1. I'm really confused with these type of problems. Can anyone give some helpful advice?

Thanks
 
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  • #2
could you say that x^4 + 1=(x^2 + i)(x^2 - i). Since this is factored into two polynomials of degree 2 the degree of the splitting field is 2*2=4?
 
  • #3
i is not a rational number. What are the roots of x^4_1? What is the smallest extension of Q that contains them?
 
  • #4
can't the roots be in an extension field? isnt' the splitting field an extension field so i would be in the extension field? The roots would be x = (-1)^(1/4) but this isn't a rational number. I think because of this the extension would have to be the complex field because no real number satisfies this. How do you know if it's the smalllest extension?
 
  • #5
buzzmath said:
can't the roots be in an extension field? isnt' the splitting field an extension field so i would be in the extension field? The roots would be x = (-1)^(1/4) but this isn't a rational number. I think because of this the extension would have to be the complex field because no real number satisfies this. How do you know if it's the smalllest extension?
Yes, that's the whole point. The splitting field is the smallest extension of Q that contains all of its roots. No, that splitting field is NOT the set of all complex numbers. What are the four roots of x4/sup]= -1?
 
  • #6
Can you write it like w = cos(360/4)+isin(360/4) then w^4 =1 so (2)^(1/2)/2+or-(2)^(1/2)/2i are the four roots so the field is Q(2^(1/2),i) but then how do you tell the degree of this field?
 
  • #7
my experience with degrees is with polynomials not an entire field
 
  • #8
Good. Yes, the splitting field is [itex]Q(\sqrt{2},i)[/itex]. The smallest field containing all rational numbers, [itex]\sqrt{2}[/itex], and i. In particular, that means it must include all numbers of the form [itex]a+ bi+ c\sqrt{2}[/itex] where a, b, c are rational numbers. Since it must be closed under multiplication, it must also include [itex]\isqrt{2}[/itex]. Since that cannot be written in the above form, it must, in fact, include numbers of the form [itex]a+ bi+ c\sqrt{2}+ di\sqrt{2}[/itex]. One can show that any number in this extension field can be written in that form. We can think of that as a vector space over the rational numbers with basis {1, i, [itex]\sqrt{2}[/itex],[itex]i\sqrt{2}[/itex]}: i.e. the vector space has dimension 4 over the rational numbers. THAT is the "degree" of the extension field: its dimension as a vector space over the rational numbers. In this case we could also have seen that by noting that [itex]\sqrt{2}[/itex] and i are both "algebraic of order 2" and that they are "algebraically independent".
 

Related to Abstract Algebra: Splitting Fields and Prime Polynomials

1. What is abstract algebra?

Abstract algebra is a branch of mathematics that deals with algebraic structures and their properties, rather than specific numbers or equations. It focuses on the study of groups, rings, fields, and other algebraic structures.

2. What are splitting fields?

Splitting fields are extensions of a given field that contain all the roots of a given polynomial. In other words, it is the smallest field in which a given polynomial can be completely factored into linear factors.

3. What are prime polynomials?

Prime polynomials are irreducible polynomials, meaning they cannot be factored into smaller polynomials with coefficients in the same field. They are important in abstract algebra because they serve as building blocks for constructing other polynomials.

4. Why are splitting fields and prime polynomials important?

Splitting fields and prime polynomials are important concepts in abstract algebra because they allow us to study and understand the behavior of polynomials over different fields. They also have practical applications in number theory, coding theory, and cryptography.

5. What are some real-world applications of abstract algebra?

Abstract algebra has numerous applications in various fields, such as physics, chemistry, computer science, and engineering. For example, group theory is used in crystallography to study the symmetry of crystals, and algebraic coding theory uses finite fields and polynomials to transmit and store data securely.

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