An Algebraist's View: Generalizing Vieta's Formula

It turns out that Vieta's formula applies to any algebraically closed field. Call a field \(F\) algebraically closed if(f) every polynomial in \(F[x]\) splits over \(F\).

Theorem: \(\mathbb{C}\) is algebraically closed.

Proof: It suffices to show that there is not a proper finite field extension of \(\mathbb{C}\). This is a healthy exercise in Galois theory, and is thus left to the reader. (Hint: Suppose, by way of contradiction, that \(L:\mathbb{C}\) is a proper finite extension of \(\mathbb{C}\). What can we say about \([L:\mathbb{C}]\)?) \(\boxed{ }\)

Thus, we see that Vieta's formula works in at least one algebraically closed field. Let's broaden the scope.

Theorem: Suppose

\(\displaystyle p(x)=\sum_{0\leq k\leq n}\lambda_kx^k \in F[x]\),

where \(F\) is an algebraically closed field, and that \(p(x)\) splits as

\(\displaystyle p(x)=\lambda_n\prod_{1\leq i\leq n}(x-\alpha_i)\).

Then, \(\displaystyle \sum_{1\leq i_1<i_2<\cdots<i_m\leq n}\alpha_{i_1}\alpha_{i_2}\cdots\alpha_{i_m}=(-1)^m\frac{\lambda_{n-m}}{\lambda_n}\).

Proof: A field is necessarily commutative and distributes over addition. Thus, the proof is simply a matter of noting the coefficients must match up. \(\boxed{ }\)

Thus, we can use Vieta's formula for arbitrary polynomials over fields.

[Edit: Sorry for the awful formatting. I'm used to writing things in pure LaTeX, so this is a little weird to me.]

Note by Jacob Erickson
5 years, 1 month ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link]( link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)


Sort by:

Top Newest

Nice bro! :D

Finn Hulse - 4 years, 9 months ago

Log in to reply

@Jacob Erickson Can you add this to a suitable skill in the Vieta Formula Wiki? Let me know if you think a different skill would be suitable.

Calvin Lin Staff - 4 years, 3 months ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...