VIPSolutionsgive answer in 2 step with explanation at the end of each step and final answer at the end: Problem A . Let ω = exp ( 2 π - 1 3 ) . Show that Q [ ω ] : = { a b ω c ω 2 | a , b , cinQ } is a field with addition and multiplication defined as in C . What is char ( Q [ ω ] ) ?Problem A. Letw = exp(25y/=1/3). Show that Qiu] := {a+hwtew? | a,b,c € Q} is a field with addition and multiplication defined as in C. What is char(Q[u])?
Question:
give answer in 2 step with explanation at the end of each step and final answer at the end:
Problem A
.
Let
ω
=
exp
(
2
π
-
1
3
)
.
Show that
Q
[
ω
]
:
=
{
a
b
ω
c
ω
2
|
a
,
b
,
cinQ
}
is a field with addition and multiplication defined as in
C
.
What is char
(
Q
[
ω
]
)
?
Problem A. Letw = exp(25y/=1/3). Show that Qiu] := {a+hwtew? | a,b,c € Q}
is a field with addition and multiplication defined as in C. What is char(Q[u])?
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Asked by: Pagal
Created at: 2025-06-03 19:19:40
Asked by: Pagal
Created at: 2025-06-03 19:19:40
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Let's solve the problem step by step:
---
**Step 1: Show that \( \mathbb{Q}[\omega] \) is a field.**
Recall:
- \( \omega = \exp(2\pi i/3) \) is a primitive third root of unity, so \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).
- Elements of \( \mathbb{Q}[\omega] \) are of the form \( a + b\omega + c\omega^2 \) for \( a,b,c \in \mathbb{Q} \).
**Addition:**
If \( x = a + b\omega + c\omega^2 \) and \( y = a' + b'\omega + c'\omega^2 \), then:
\[
x+y = (a+a') + (b+b')\omega + (c+c')\omega^2
\]
which is clearly in \( \mathbb{Q}[\omega] \).
**Multiplication:**
Use the relations \( \omega^3 = 1 \) and \( 1 + \omega + \omega^2 = 0 \).
Check closure under multiplication:
\[
x \cdot y = (a + b\omega + c\omega^2)(a' + b'\omega + c'\omega^2)
\]
Expand and combine like terms using \( \omega^3 = 1 \) and \( \omega^2 = -1 - \omega \).
Thus, the product is again a linear combination of \( 1, \omega, \omega^2 \) with rational coefficients.
**Existence of Inverses:**
\( \omega \) satisfies \( x^2 + x + 1 = 0 \), so it is a root of an irreducible polynomial over \( \mathbb{Q} \).
Thus, \( \mathbb{Q}[\omega] \) is a field (it is the splitting field of \( x^3-1 \) over \( \mathbb{Q} \)), and every nonzero element has an inverse.
**End of Step 1 Explanation:**
We have checked that \( \mathbb{Q}[\omega] \) is closed under addition and multiplication, and that every nonzero element has an inverse, so it is a field.
---
**Step 2: Determine the characteristic of \( \mathbb{Q}[\omega] \).**
Recall:
- The characteristic of a field \( F \) is the smallest positive integer \( n \) such that \( n\cdot 1 = 0 \). If no such \( n \) exists, the characteristic is 0.
Since \( \mathbb{Q}[\omega] \) contains \( \mathbb{Q} \), and \( \mathbb{Q} \) has characteristic 0, so does \( \mathbb{Q}[\omega] \).
**End of Step 2 Explanation:**
Because \( \mathbb{Q}[\omega] \) contains a copy of \( \mathbb{Q} \), its characteristic must be 0.
---
**Final Answer:**
\[
\mathbb{Q}[\omega] = \{a + b\omega + c\omega^2 \mid a,b,c \in \mathbb{Q}\}
\]
is a field, and its characteristic is \( \boxed{0} \).
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